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University of Groningen
Electromagnetic pulse propagation in one-dimensional photonic crystalsUitham, Rudolf
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Electromagnetic pulse propagation in
one-dimensional photonic crystals
RUDOLF UITHAM
Zernike Institute Ph.D. thesis series 2008-23
ISSN 1570-1530
The work described in this thesis was performed in the research group Theory of
Condensed Matter of the Zernike Institute for Advanced Materials at the University
of Groningen. This work is financially supported by NanoNed, a national nanotech-
nology programme coordinated by the Dutch Ministry of Economic Affairs.
Printed by GrafiMedia, University Services Department, University of Groningen,
Blauwborgje 8, 9747 AC, Groningen, The Netherlands.
Copyright c© 2008 Rudolf Uitham.
RIJKSUNIVERSITEIT GRONINGEN
Electromagnetic pulse propagation inone-dimensional photonic crystals
Proefschrift
ter verkrijging van het doctoraat in de
Wiskunde en Natuurwetenschappen
aan de Rijksuniversiteit Groningen
op gezag van de
Rector Magnificus, dr. F. Zwarts,
in het openbaar te verdedigen op
vrijdag 5 december 2008
om 13:15 uur
door
Rudolf Uitham
geboren op 18 april 1977
te Delfzijl
Promotor: Prof. dr. J. Knoester
Copromotor: Dr. B. J. Hoenders
Beoordelingscommissie: Prof. dr. H. A. de Raedt
Prof. dr. H. P. Urbach
Prof. dr. A. T. Friberg
ISBN: 978-90-367-3633-6
Contents
1 Introduction 1
1.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Applications of photonic crystals . . . . . . . . . . . . . . . . . . . 6
1.4 Metallodielectric photonic crystals . . . . . . . . . . . . . . . . . . 8
1.5 Fundamental physics in photonic crystals . . . . . . . . . . . . . . 8
1.6 Theoretical research . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Precursors in homogeneous media . . . . . . . . . . . . . . . . . . 10
1.8 Precursors in photonic crystals . . . . . . . . . . . . . . . . . . . . 11
1.9 Transmission coefficient from a sum over all light-paths . . . . . . . 13
1.10 Scattering in the absence of one-to-one coupling of field modes . . . 14
1.11 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 The Sommerfeld precursor in photonic crystals 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Model for the one-dimensional photonic crystal . . . . . . . . . . . 21
2.3 Applied pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Plane-wave transmission coefficient for the multilayer . . . . . . . . 25
2.5 Wavefront of the transmitted pulse . . . . . . . . . . . . . . . . . . 26
2.6 Sommerfeld precursor . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 The Brillouin precursor in photonic crystals 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Model for the photonic crystal . . . . . . . . . . . . . . . . . . . . 37
vi Contents
3.3 Transmission coefficient of the photonic crystal . . . . . . . . . . . 39
3.4 Transmittance of the photonic crystal . . . . . . . . . . . . . . . . . 43
3.5 Investigation of Brillouin precursor with steepest descent method . . 45
3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Multilayer transmission coefficient from a sum of light-rays 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Model for the medium . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Electromagnetic field in the medium . . . . . . . . . . . . . . . . . 57
4.4 Path decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Path realizations for multiply-scattered, transmitted
light-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Transmission coefficient via sum of all possible paths . . . . . . . . 67
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Scattering from systems that do not display one-to-one coupling of modes 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Hybrid mode expansions . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Modes in the rotated multilayer slab . . . . . . . . . . . . . 81
5.2.2 Scattering from a semi-infinite line . . . . . . . . . . . . . . 84
5.2.3 Scattering from a layer with finite width . . . . . . . . . . . 88
5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Summary and Outlook 99
A Accuracy of the calculation of the Sommerfeld precursor 103
B Method of steepest descent 107
C Derivation of hybrid completeness relations 111
C.1 The special eigenfunction expansions . . . . . . . . . . . . . . . . 111
C.2 Transformation of series . . . . . . . . . . . . . . . . . . . . . . . 113
C.3 Expansion of a plane wave into the free space modes . . . . . . . . 115
D List of publications 117
Contents vii
Bibliography 119
Nederlandse Samenvatting 127
Acknowledgements 133
Chapter 1
Introduction
1.1 Photonic crystals
The fastest exchange of information is achieved by mediating it with electromagnetic
pulses. When these pulses can be controlled with low loss of energy and within a
small spatial volume, efficient optical devices are within reach. During the last twenty
years, much progress has been made in the development of photonic crystals [1]
since it is expected that with these manmade structures the low-loss and small-scale
manipulation of light can be realized.
Photonic crystals are composite materials in which the building blocks of the
crystal unit cell are dielectric1 media. In photonic crystals, the index of refraction
varies periodically in space, where the period is given by the spatial extent of the
unit cell. The dimension of the photonic crystal is given by the number of indepen-
dent spatial directions along which the refractive index varies repeatedly. Example
models for a one-, two- and three-dimensional photonic crystal have been depicted in
Fig. 1.1. An electromagnetic field that impinges upon a photonic crystal is reflected
periodically inside the medium, where the reflectance of each unit cell increases with
increasing contrast of the refractive indices of the constituents. When, for applied
harmonic plane waves that propagate in a given direction, the wavelength and the
crystal period along the propagation direction compare such that the back-reflected
waves interfere constructively, the field will be strongly rejected from the crystal. In
this case, the amplitude of the field within the crystal decays exponentially with the
1Photonic crystals can also be constructed from both dielectric and metallic components. These
metallodielectric photonic crystals are discussed below in a separate paragraph.
2 Introduction
(a) (b) (c)
Figure 1.1: Example models for (a) a one-dimensional photonic crystal, (b) a two-
dimensional photonic crystal and (c) a three-dimensional photonic crystal. In regions
with different colors, the index of refraction takes on different values.
distance to the boundary surface of the medium and there is no traveling field inside
the medium. The range of wavelengths or, equivalently, frequencies, for which no
propagating wave solutions exist in the crystal is called the photonic band gap. If
the photonic band gap extends to all possible propagation directions of the field, it is
complete. The counterpart of field rejection from the crystal occurs when the wave-
length and crystal period along the propagation direction of the incident harmonic
plane waves compare such that the back-reflected waves from unit cells interfere de-
structively. In that case, the field will be well-transmitted through the crystal. The
photonic crystal thus effects a selectivity in the reflection and transmission of electro-
magnetic waves, where the selection criterion is the wave-vector of the field. Fig. 1.2
depicts a sketch of the dispersion relation for electromagnetic harmonic plane waves
in a uniform one-dimensional homogeneous medium and in a one-dimensional pho-
tonic crystal. The effects of the inhomogeneities of the medium are seen in a splitting
of the bands, the frequency solutions ω that correspond to the real wave number k, at
the edges of the Brillouin zone at k = ±π/l, resulting in the photonic band gap.
Electromagnetic waves in photonic crystals have a strong analogy in the field of
solid state physics. This similarity is given by electrons in interaction with a crystal
lattice of atoms or molecules, where the crystal represents a periodic potential for the
electrons. An electronic band gap results if the electrons are Bragg diffracted [2].
The charge configuration of the atoms or molecules and the structure of the crystal
together determine the conduction properties of the medium. In photonic crystals,
the index of refraction of the material components and the crystal structure both de-
termine the dispersion of light. The differences are that the electromagnetic wave
has a polarization and satisfies the Maxwell equations whereas the electron wave is a
1.1 Photonic crystals 3
k
ω
π/l−π/l
(a)
photonic band gap
photonic band gap
k
ω
π/l−π/l
l
k
(b)
Figure 1.2: Dispersion relation for electromagnetic plane waves in (a) a one-
dimensional homogeneous medium with artificial period l and (b) a one-dimensional
photonic crystal with period l. The splitting of the degenerate frequency solutions at
the Brillouin zone edges leads to the formation of photonic band gaps.
scalar field that obeys Schrodinger’s equation [3].
In conclusion, the unusual dispersion relation of electromagnetic waves in pho-
tonic crystals and in particular the possible presence of a photonic band gap render
photonic crystals useful for manipulating the propagation of light. A legitimate ques-
tion that could arise at this point is: why use such complex materials as photonic
crystals and why not simply use the reflective properties of metals or the phenomenon
of total internal reflection in dielectrics to prevent light from going somewhere? The
answer lies within the energy losses and the scale. Photonic crystals are much less
dissipative than metal mirrors for the control of electromagnetic waves [4]. Further,
as compared to total internal reflection based waveguides such as for instance glass
fibre cables, photonic crystals can manipulate the flow of light at a much smaller
scale, namely that of the wavelength of the guided light itself [1].
4 Introduction
1.2 Historical overview
The earliest study of electromagnetic wave propagation in periodic media dates from
the year 1888, when Lord Rayleigh studied the peculiar reflective properties of a
crystalline mineral [5]. Rayleigh tried to explain ”the high degree of selection (of
wave-vectors) and copiousness of the reflection of the crystal”. Not convinced by the
earlier explanation that the peculiar reflection was caused by the presence of a single
narrow layer inside the crystal, then called ”the seat of color”, Rayleigh successfully
assumed that the mineral consisted of a large number of alternating layers. The min-
eral was thus identified as a natural periodic multilayer or one-dimensional photonic
crystal, see Fig. 1.1(a).
Photonic crystals have also been found elsewhere in nature, though not very
abundantly. Iridescent colors due to crystalline medium structures are for instance
observed in the wings of certain butterflies [6] and in opals. The known natural pho-
tonic crystals do not have a complete band gap, which is required for many of the
proposed devices to operate. Photonic crystals with a complete photonic band gap
can only be realized by means of artificial fabrication. In the following paragraphs,
the historical overview of the fabrication of photonic crystals will be given.
The periodic multilayer has been studied intensively throughout the twentieth
century [7] and nowadays it has several applications, for instance as anti-reflection
coating, which was first developed by Smakula [8] in 1936, as Fabry-Perot filter and
in distributed feedback lasers [9]. The generalization of the periodic multilayer to
media with periodicity in more than one dimension was proposed by Bykov [10, 11]
in 1972, with the idea to inhibit spontaneous optical emission. Then, in the 1980s,
Yablonovitch [12] recognized that the photonic crystal could be used to increase
the performance of lasers and other devices that suffer from the previously uncon-
trollable spontaneous emission. Another important application, namely using pho-
tonic crystals as a mechanism for the localization of light, was proposed in 1987 by
John [13]. These propositions strongly provoked the interest in photonic crystals and
thus marked the birth of the photonic crystal area.
Inspired by the theoretical ideas of John [13] and Yablonovitch [12], the challenge
arose to fabricate a photonic crystal with an actual complete band gap in the visible
part of the frequency spectrum. At that time, in the 1980s, it was experimentally still
impossible to create highly regular three-dimensional lattices with periodicity be-
low the micrometer, so the fabrication of a photonic crystal that could influence the
1.2 Historical overview 5
flow of visible light2 was still beyond reach. The first attempts to create a complete
band gap material therefore involved crystals with much larger periodicities. The first
three-dimensional photonic crystal, with periodicity at the millimeter scale, was fab-
ricated by Yablonovitch and Gmitter [15]. This crystal was an array of spherical voids
filled with air in a matrix of dielectric material, arranged in a face-centered cubic lat-
tice. Their attempt to find a complete photonic band gap was unfruitful; the measured
transmission spectra did not show a complete gap. This was quite remarkable be-
cause contemporary simulations, which used the scalar wave approximation [16], did
predict a complete band gap for the face-centered cubic lattice. Later simulations,
which incorporated the vectorial nature of the electromagnetic field [17, 18], con-
firmed the absence of a complete band gap for Yablonovitch and Gmitter’s crystal.
Subsequently, Ho et al. [19] suggested using a diamond structure, for which the latest
simulations did promise a complete band gap. Consequently, Yablonovitch created
the diamond structure by drilling cylindrical holes in a dielectric material and it was
for this crystal that the transmission spectra revealed the first complete band gap [20].
After Yablonovitch’ first experimentally realized diamond structure crystal with
a complete band gap, different structures with band gaps were proposed and realized,
such as the woodpile structure [21, 22]. Experimental measurements on woodpile
structure based photonic crystals have also been intensively performed by the group
of A. Polman [23]. The woodpile structure has the advantage that it can be fabricated
layer-by-layer which facilitates further engineering of the interior of the crystal. Af-
ter a long route of downsizing the crystal period by using increasingly advanced pro-
cessing techniques [22,24–27], three-dimensional periodicity at the micrometer scale
was reached by adoption of a crystallization process from nature. As it was known
that natural opals illuminated with white light reflect colored light where the color
varies with the angle of reflection, in other words these crystals were known to have
a band gap, the community started to fabricate artificial opals by copying the natural
process of colloidal self-assembly of monodisperse spheres [28]. The group of A.
van Blaaderen proposed such a self-assembly route for photonic crystals with a band
gap in the visible region [29]. Experimental measurements of optical properties of
synthetic opals from monodisperse polystyrene colloids [30–35] showed band gaps
at wavelengths comparable to the diameter of the spheres. A severe disadvantage
of self-assembling crystals is that it is difficult to engineer them to fulfil particular
applications. For instance, it is hard to control the introduction of defects into the
structure. Therefore, Joannopoulos and coworkers [36–38] returned to planar struc-
2The spectrum of vacuum wavelengths of visible light ranges roughly from 400nm to 700nm [14].
6 Introduction
tures with periodicity in two dimensions and experimentally showed the presence of
the photonic band gap. The ability to guide visible light in two dimensions through
small channels around sharp bends with very low loss was predicted [39] and both
numerically [40] and experimentally [41] verified. The dependence of the coupling
of light from an external point source to a three-dimensional photonic crystal on the
relative position of the light source with respect to the crystal lattice has been spatially
resolved beyond the dimensions of the unit cell with a near-field scanning microscope
by the group of L. Kuipers [42].
1.3 Applications of photonic crystals
There are numerous applications of photonic crystals. In 1994, Meade et al. [39]
first proposed using them as waveguides. A waveguide is obtained from a photonic
crystal by introducing a line of defects in it, this has been illustrated in Fig. 1.3(a).
Since the light cannot continue its propagation in the perfect part of the crystal, it is
forced to follow the defect route along which the periodicity is broken, even if this
line has sharp bends. Although the light does not escape the photonic crystal waveg-
uide at bends, part of the light undergoes back-reflection there, which also results in
transmission loss. Much effort has been spent to reduce these back-reflection losses,
for instance by rearranging the lattice near the bend [43], smoothing the bend and
changing locally the width of the guide [44] and adding appropriate defects at the
bend corners [45].
Confinement of the light to the waveguide that is independent of the shape of the
guide can not be achieved in waveguides that are based on total internal reflection,
where there exists a minimal bend radius below which the light escapes from the
waveguide. For the guiding of for instance telecom waves (wavelength 1.5µm in
vacuum [46]) in a glass fibre cable surrounded by air, the bend radius, which is the
outer radius of the circularly bent cable, should be at least a few millimeters. As
compared to a photonic crystal waveguide, which has extensions of the order of the
wavelength of the guided light, this is a significant difference in size. This explains
why photonic crystals can manipulate the flow of light at small scale.
If instead of a line of defects, only a single point defect is introduced in the
photonic crystal, as Yablonovitch and Gmitter [47] first proposed in 1991, local elec-
tromagnetic modes can exist with frequencies that lie inside the photonic band gap.
Thus, photonic crystals can be utilized as microcavities, which are essential compo-
nents of lasers and filters. The photonic crystal microcavity has been illustrated in
1.3 Applications of photonic crystals 7
(a) (b)
Figure 1.3: Photonic crystals with defects. Different colors indicate regions with
different indices of refraction, the defect building blocks have been given the darkest
color. (a) line of defects, resulting in a waveguide and (b) point defect, resulting in a
microcavity.
Fig. 1.3(b). For a good performance, it is required that the cavity has a high qual-
ity factor and a small mode volume. A high quality factor means low energy loss
per radiation cycle which implies having a well-defined frequency and a small mode
volume ensures high coherence. Since, with photonic crystal surroundings, the local
electromagnetic modes in the cavity are confined with low loss, the quality factor of
such a cavity can reach high values of over ten thousand [48]. Moreover, the size of
the cavity can be brought down to the order of the wavelength, which implies a rather
small mode volume for the cavity. Various methods have been proposed to further in-
crease the quality factor and decrease the mode volume, as for instance by adjusting
the cavity geometry [49] and recycling the radiated field [50], [51]. The first working
pulsed laser based on a photonic crystal microcavity was reported in 1999 by Lee et
al. [52].
Further proposed photonic crystal applications are beam splitters [53], add/drop
filters [54], switches [55,56], waveguide branches [57], transistors [58], limiters [59,
60], modulators [61–65], amplifiers [66, 67] and optical delay lines [68]. Many pho-
tonic crystal applications have been realized with good performance such as the drop
filter [69], optical filter [54], polarization splitter [70], Y-splitter [71–73] and Mach-
Zehnder interferometer [74].
8 Introduction
1.4 Metallodielectric photonic crystals
The control of electromagnetic microwaves can not only be realized with dielec-
tric photonic crystals but also, and even better, with metallodielectric photonic crys-
tals [75], which have both metal and dielectric components. For pure dielectric crys-
tals, a significant fraction of the electromagnetic field penetrates through a unit cell so
that several of these cells are needed to achieve Bragg scattering [4]. For microwaves,
an interface between a dielectric and a metal medium reflects the field much more ef-
ficiently3 than an interface between two dielectric media. With the introduction of
metallic components, photonic band gaps for microwaves can therefore be realized
with less unit cells. Besides having the advantage of efficient reflection, the use of
metallic components also introduces additional functional properties to the crystal.
For instance, each cell can be designed with a circuit element having adjustable in-
ductance and capacitance. These enriched photonic crystals can be used, for example,
to increase the performance of microwave antennas [75–77] or reduce the backwards
radiation of cell phones [75].
1.5 Fundamental physics in photonic crystals
Until the photonic crystal area, it has always been assumed that the spontaneous pho-
ton emission rate of an atom or molecule could not be influenced. However, as has
been mentioned earlier, Bykov [10, 11] first put forward the idea that these emission
rates could possibly be altered with photonic crystals. Experimental verification for
the control of the spontaneous emission of quantum dots by three-dimensional pho-
tonic crystals has been given for instance by the group of W. L. Vos [78, 79]. Not
only the inhibition of spontaneous atomic emission of photons, but also various other
interesting fundamental physics phenomena have been observed in photonic crystals.
Ozbay et al. [80] and Bayindir et al. [81–83] theoretically and experimentally demon-
strated that photons can hop from one to another nearby cavity because of a coupling
between both cavity modes. They found that this hopping could be described with
the tight-binding method and observed high transmittance of electromagnetic waves
through a sequence of microcavities. Kosaka et al. [84] investigated the nonlinear
3For an efficient reflection, the metal parts in a metallodielectric photonic crystal should be included
in each unit cell as isolated components, since otherwise long-range electric currents are induced by the
field and these would cause significant energy losses in the photonic crystal [4]. For this reason, the
metallic mirror is relatively dissipative in the manipulation of electromagnetic waves.
1.6 Theoretical research 9
optical phenomenon of superprism in photonic crystals, demonstrated that photonic
crystals can have a negative index of refraction, which was utilized for applications
such as beam steering [85], spot size conversion [86,87] and self-collimation [88–91].
1.6 Theoretical research
The numerous applications of photonic crystals and the interesting fundamental physics
phenomena that can be observed in photonic crystals strongly ask for a good compre-
hension of these materials and motivate the search for solutions to the many unsolved
problems that remain in the theory behind electromagnetic wave propagation in pho-
tonic crystals. To give an impression of the sort of problems that exist, we mention a
few of these remaining questions. It has been calculated and experimentally observed
that the group velocity of electromagnetic waves in photonic crystals can become ex-
tremely small for frequencies at the edge of the photonic band gap [92, 93]. It is
not clear what the physical meaning of this vanishing group velocity is; does it im-
ply a vanishing signal velocity? Another remaining challenge is to extrapolate the
theory of partial coherence [7] from homogeneous media to photonic crystal materi-
als. Apart from interesting fundamental issues that emerge with the extension of this
theory, the elaboration is of practical relevance because of the previously mentioned
demand for high coherence of for instance photonic crystal based lasers.
Although the exact theory of electromagnetic wave propagation in photonic crys-
tals is at hand in the form of Maxwell’s equations, the characteristic multiple scat-
tering of light within these materials makes the detailed analytic description of the
propagation a complicated task. As a consequence, one is more or less forced to
solve the Maxwell equations numerically if one wants to calculate the amplitude of
a pulse after it has had some interaction with the photonic crystal. In this thesis,
however, it will be shown that some phenomena that come with pulse propagation
in photonic crystals can still be fully described with transparent analytic expressions,
where transparent analytic expressions are simple functions of the input pulse and
material parameters.
The various phenomena belonging to pulse propagation in photonic crystals that
are investigated with (semi-)analytic methods in this thesis are listed in this para-
graph. First, the so-called Sommerfeld precursor [94] field, here calculated for an
electromagnetic pulse that has been transmitted through a one-dimensional photonic
crystal, is obtained in a closed form. Thereafter, the Brillouin precursor [94], also
calculated for an electromagnetic pulse that has been transmitted through a one-
10 Introduction
dimensional photonic crystal here, is obtained semi-analytically. This means that
an analytic expression for the Brillouin precursor is obtained, but the visualization of
the amplitude of this precursor is realized with the use of numerical methods. Af-
ter the precursor part of this thesis, a transparent analytic expression is derived for
the transmission coefficient of the multilayer. Written in this alternative form, each
term of the transmission coefficient directly represents a transmitted light-ray. The
last part of this thesis treats electromagnetic wave scattering from objects for which
there is no one-to-one coupling of the natural modes of the field inside and outside
the object. Two sets of electromagnetic modes are established, one for the fields in-
terior and one for the field exterior to the scatterer, such that these modes together
yield a hybrid completeness relation. With this relation, it turns out to be possible to
calculate the scattered fields. The various concepts that are encountered in this thesis,
such as precursors, are introduced in the remainder of this chapter.
1.7 Precursors in homogeneous media
Electromagnetic pulse propagation in linear isotropic homogeneous dielectric media
with frequency dispersion and absorption has been thoroughly studied in a classi-
cal paper of Sommerfeld and Brillouin [94]. This theoretical work originates from
1914 and, though old, it is still considered as a milestone in electromagnetic wave
propagation. In the course of time, Sommerfeld and Brillouin’s theory has been fur-
ther refined by Oughstun and Sherman [95]. Substantial part of Sommerfeld and
Brillouin’s analysis of pulse propagation in homogeneous media is devoted to their
theoretical discovery of precursors.
The precursors, which are named after their discoverers as the Sommerfeld and
the Brillouin precursor [14], are distinct wave patterns with usually very small am-
plitudes and high frequencies as compared to the applied (optical) pulse. The wave
patterns, the characteristics in the behavior of the electromagnetic field as a function
of time, of forerunners are rather universal, quite independent of the exact shape of
the incident pulse and the exact values of the medium parameters. As a consequence
of the frequency dispersion and absorption in the homogeneous medium, each fre-
quency component that is provided by the applied pulse propagates at its own speed
and attenuates with its own decay constant. The precursors arise within the medium
as a consequence of the very complicated interplay of various frequency components
and are related with the dispersion characteristics of the waves within the medium.
The forerunners are composed of those frequency components of the applied
1.8 Precursors in photonic crystals 11
pulse that have a relatively weak interaction with the medium. The weakly interacting
frequency components are those that lie in regions with small absorption, far away
from the resonances of the medium. This explains the maximum number of possible
precursors that can arise in materials that are modeled as Lorentz media [96, 97]. In
a single electron resonance Lorentz medium, two precursors can arise [96]: one with
frequencies much higher than the resonance, this forerunner is called the Sommerfeld
precursor, and one with frequencies far below the resonance, the Brillouin precursor.
In a multiple electron resonance Lorentz medium, the maximum number of precur-
sors that can arise is equal to the number of off-resonance regions in the frequency
spectrum of the medium, where the interaction with the electromagnetic field is rela-
tively weak. It depends on the values of the medium parameters, whether all of these
precursors will actually appear [97]. Therefore, the frequency components that have
a relatively weak interaction with the medium do not necessarily produce a precursor.
In a single relaxation Debye model medium, for instance, it has been calculated that
only the low-frequency Brillouin precursor arises [98,99], although the interaction of
the field with the medium is weak at high frequencies as well.
Apart from the fact that the precursors form an intrinsic part of the transmitted
field in many dispersive media, the forerunners also have an interesting property that
makes an extension of their study to inhomogeneous media worthwhile. For ho-
mogeneous media, it has been shown that the peak amplitudes of precursors do not
decay exponentially with propagation distance but algebraically [100]. Their deep
penetration capability turns the precursors into candidate signals for medical imag-
ing and underwater communication [101]. It is therefore interesting to find out how
the peak amplitude decays in inhomogeneous media. Although we do not answer
this question, in this thesis we will lay the groundwork for such a calculation. The
first direct experimental observation of precursors was reported in 1969 by Pleshko
and Palocz [102] for microwaves in a dispersive waveguide. Optical precursors have
been observed in GaAs [103], CuCl [104] and in water [100].
1.8 Precursors in photonic crystals
Throughout this thesis, the photonic crystals under investigation have the simplest
possible geometry, namely that of the stratified, periodic multilayer. The inhomo-
geneity of the photonic crystal gives rise to the presence of what is called waveguide
dispersion, which is a frequency-dependent response as a result from the geometry
of the medium. In order to present realistic photonic crystals in this thesis, the slabs
12 Introduction
of the crystal are also provided with frequency dispersion and absorption, so in total
there are two different origins of the dispersion and absorption. The frequency dis-
persion and absorption of each dielectric slab is modeled as that of a single-electron
resonance Lorentz medium [105], exactly as the homogeneous medium that was con-
sidered in Sommerfeld and Brillouin’s analysis [94]. The aim of this research is to
determine how the precursors are affected by the inhomogeneities of the medium.
Stated in other words, the purpose of this study is to reveal the interplay of frequency
and waveguide dispersion and absorption in our medium with respect to the formation
of the precursors.
In our analysis of pulse propagation in photonic crystals, the line of Sommerfeld
and Brillouin’s research [94, 96] on pulse propagation in homogeneous media is fol-
lowed. The pulse under consideration is incident onto the photonic crystal from one
side and after transmission it is evaluated as a function of the medium parameters
and time. The photonic crystal is surrounded by vacuum, so that only the effects of
the crystal are pointed out. The analysis of the transmitted pulse is carried out from
its wavefront along the first (Sommerfeld) precursor up to and including the second
(Brillouin) precursor. Asymptotic analysis is applied to the Fourier integral repre-
sentation of the transmitted pulse to obtain information about the wavefront and the
Sommerfeld precursor. The Brillouin precursor is analyzed with the more rigorous
method of steepest descent. This analysis is supported with plots of the transmittance
of the medium; these plots clearly show the dominant contributions to the transmitted
field at successive instants of time.
The following results are obtained for pulse propagation in our photonic crystal.
The wavefront of the pulse propagates at the speed of light in vacuum, as it does in ho-
mogeneous media [94]. The shape of the Sommerfeld precursor is not altered by the
medium inhomogeneities since it merely experiences the spatial average of the pho-
tonic crystal medium. The Brillouin precursor, however, can be severely distorted by
the inhomogeneities of the medium. This distortion depends in a complicated manner
on the medium parameters. As one would predict intuitively, the Brillouin precursor
is increasingly distorted with augmented index contrast. It is clearly exposed in the
transmittance plots that, after a rise time of the amplitude of the Brillouin precursor,
the frequencies of the dominant stationary point contributions to the field approach
those of the scattering resonances of the medium.
After the calculation of the precursors in the one-dimensional photonic crystal,
the attention is focused on the light when it is inside the crystal medium, with the
purpose of gaining deeper insight in the characteristics of the transmitted field.
1.9 Transmission coefficient from a sum over all light-paths 13
1.9 Transmission coefficient from a sum over all light-paths
A key ingredient in the description of propagation of electromagnetic waves through
multilayer media is the transmission coefficient, which is the ratio of the amplitude
of the transmitted electric field to that of the incident electric field. Usually, the trans-
mission coefficient is calculated via the transfer matrix method [106], which relates
the fields in subsequent layers by demanding continuity of the tangential components
of the total electric and magnetic fields at the interfaces. It is also interesting to derive
the transmission coefficient by summing the amplitude coefficients of all possible in-
dividual transmitted light-rays. With such a derivation of the transmission coefficient
it can be expected that the resonance structure of the medium is displayed better,
since resonances are constructive interferences of light-rays and these light-rays are
obscured in the transfer matrix derivation. It can be expected that the sum-over-all-
light-rays derivation brings the transmission coefficient in a very simple, natural, or
elementary form. This prediction turns out to be true.
For a monolayer, the sum over all light-paths is rather easily obtained [106], be-
cause the only possibility for the light is to scatter back and forth a number of times
between the two interfaces of the medium. The sum of all the transmitted light-rays
is then immediately identified as a geometric series. For the case of a medium with
more than one layer, it first seemed that the sum of all transmitted light-rays could not
so easily be obtained because of a dramatic increase of the number of possible paths
for the transmitted light. However, this problem is successfully solved and it is even
possible to immediately write down by hand the transmission coefficient in the alter-
native form for a multilayer with a few slabs. To our knowledge, this has never been
achieved before. The starting point in the derivation is to find a basis for the indi-
vidual transmitted light-rays, a basis from which all possible intermediate reflections
against interfaces within the multilayer medium can be obtained. It turns out that the
elements of the basis for the reflections, taken together with the accompanied extra
propagation path elements of the light inside the medium, can be chosen as loops. A
loop is the closed path that corresponds to a back-forth scattering between a pair of
interfaces. In the above mentioned monolayer, the path that corresponds to a single
back-forth scattering of the light-ray between the two interfaces is such a loop.
With the requirement that the loops between the various different interfaces of
the multilayer should somehow be treated on an equal footing, all possible transmit-
ted light-rays through the multilayer are exactly reproduced with a geometric series
of which the argument is multilinear in the different loops. The exact combinatorics
14 Introduction
of the various loops follows directly from demanding continuity of the light-paths.
Thus, a set of rules is derived with which the transmission coefficient of any multi-
layer medium can immediately be written down in terms of Fresnel coefficients [14]
and slab-propagation factors. It is to be expected that this can be done as well for the
reflection coefficient of the multilayer.
1.10 Scattering in the absence of one-to-one coupling of field
modes
The scattering of electromagnetic waves has been investigated for a wide variety of
scattering objects. Classic examples are Sommerfeld’s study on the deflection of
light at the edge of an infinitely thin conducting sheet [107] and Lord Rayleigh’s
analysis of the effect of scratches in a conducting plane, modeled as semi-cylindrical
excrescences, on the polarization of the reflected light [108].
The natural modes of the electromagnetic field in a given part of space, for in-
stance inside a homogeneous scattering object, are the solutions of the vectorial wave
equation that the satisfy boundary conditions for the field in that region. When the
natural modes of the electromagnetic field in- and outside a scattering object cou-
ple one-to-one, the scattered fields are obtained from equating the field amplitudes
per mode. One-to-one coupling between the interior and exterior natural modes of a
scattering object takes place if both modes are similar along the object’s boundary,
which is only the case for a few scattering objects with simple geometries. When
the internal and external natural modes of the scatterer are dissimilar, one external
natural mode generally couples to an infinite number of interior natural modes and
vice versa. With this mismatch between the two sets of natural modes, a calculation
of the scattered fields thus generally results in an infinite set of equations, where each
equation relates the incident, transmitted and reflected field amplitudes at a different
mode couple.
Quite surprisingly, there is a way out to the problem of lacking one-to-one cou-
pling of the in- and exterior electromagnetic natural modes of a scatterer. With spe-
cific conditions for the electromagnetic fields in- and outside of the medium, two sets
of modified natural modes are obtained. With these sets of modes, a hybrid complete-
ness relation is established that is bilinear in the in- and exterior modes, where the
adjective hybrid merely indicates that both the in- and exterior modes are involved.
With this relation and after some manipulation, it is possible to expand all fields into
either set of modes so that the scattered field amplitudes again follow per mode.
1.10 Scattering in the absence of one-to-one coupling of field modes 15
The last part of this thesis treats electromagnetic wave scattering from objects of
which the internal natural modes do not couple one-to-one with the external ones.
As mentioned before, this means that the electromagnetic modes in the interior and
exterior of the scattering object are dissimilar and the novel feature of our electro-
magnetic wave scattering theory is that the spatial variation of the index of refraction
inside the scattering object is allowed to differ from that outside the scatterer. This
is of particular relevance for the coupling of light into multi-dimensional photonic
crystals, since these crystals have inhomogeneous boundaries only.
For the theory to be applicable, the spatial variation of the index of refraction
should admit for a separation of the vectorial wave equation. The vectorial wave
equation follows immediately from Maxwell’s equations. Examples of object geome-
tries that allow for a separated vectorial wave equation have been depicted in Fig. 1.4.
For the two-dimensional insect eye, depicted in Fig. 1.4(a), and for transverse elec-
(a) (b)
Figure 1.4: Model examples of objects in which the vectorial wave equation sepa-
rates: (a) the two-dimensional insect eye and (b) the telegrapher surface. Different
colors indicate regions in which the refractive index can take on different values. The
light is supposed to enter from above.
tric polarization, the vectorial wave equation separates in a two-dimensional spherical
coordinate system. As a consequence of the tangential variation of the index of re-
fraction inside the insect eye, the interior natural modes do not resemble the exterior
natural modes and there is no one-to-one coupling between these modes. For the
telegrapher surface, depicted in Fig. 1.4(b), the vectorial wave equation separates in
a rectangular coordinate system. Here, the interior and exterior modes do not couple
one-to-one because of the variation of the index of refraction in the horizontal direc-
tion inside the object. As mentioned, the analytic calculation of the scattered fields
from this sort of objects, that lack the one-to-one coupling of interior and exterior
electromagnetic natural modes, is still possible. The central element in this theory,
16 Introduction
the hybrid completeness relation, was taken from an analysis of E. Hilb on mode
expansions generated by inhomogeneous differential equations [109].
Hilb showed the following. Consider, on the one hand, the modes generated by
a homogeneous differential equation with a given set of boundary conditions and, on
the other hand, the modes generated by an inhomogeneous differential equation with
a different set of boundary conditions. When both sets of boundary conditions are
properly chosen, Hilb proved that both sets of modes lead to a hybrid completeness
relation and that the modes are biorthogonal if the source term of the inhomogeneous
equation satisfies certain conditions [109].
In the last part of this thesis, Hilb’s theory is applied to the modes of the electro-
magnetic field in- and outside of a rotated multilayer, depicted in Fig. 1.5(b). When
Appliedfield
(a)
Appliedfield
(b)
Figure 1.5: The multilayer in two different orientations with respect to the applied
field. (a) conventional orientation: the applied field enters the medium through a
homogeneous boundary. (b) rotated over 90 degrees: the field is incident on an in-
homogeneous boundary surface. Different colors indicate regions that have different
indices of refraction.
the multilayer has its conventional orientation with respect to the incident field, as
depicted in Fig. 1.5(a), the entrance plane separates two homogeneous spaces and the
internal and external natural modes are similar along this edge. If the multilayer is
rotated over 90 degrees, as depicted in Fig. 1.5(b), one of the inhomogeneous edges
of the medium becomes the entrance plane and the interior and exterior natural modes
along this edge are not similar.
In the homogeneous part of space outside of the scatterer, the modes satisfy a
homogeneous differential equation, the Helmholtz equation, and inside the medium
the modes fulfil an inhomogeneous version of the Helmholtz equation, where the
1.11 Outline of this thesis 17
inhomogeneity is effectively a driving force which arises from the induced polariza-
tion. With the use of the hybrid completeness relation, the amplitudes of the reflected
and transmitted fields along the inhomogeneous interface of the rotated multilayer
medium are calculated.
1.11 Outline of this thesis
The outline of this thesis is as follows. Globally, three separate aspects of elec-
tromagnetic wave propagation in photonic crystals are treated and as such the the-
sis can be divided into three parts. In the first part, which involves the chapters 2
and 3, the transmitted Sommerfeld and Brillouin precursors are calculated in the
one-dimensional multilayer. The second part, chapter 4, is devoted to the alterna-
tive formulation of the transmission coefficient. The third and last part, that involves
chapter 5, treats the scattering theory of electromagnetic waves against objects of
which the interior natural modes of the electromagnetic field do not couple one-to-
one to the exterior ones. A brief summary and outlook is given in chapter 6.
Chapter 2
The Sommerfeld precursor in
photonic crystals
In this chapter1, we calculate the Sommerfeld precursor of an electromagnetic pulse
that has been transmitted through a stratified one-dimensional photonic crystal. The
photonic crystal slabs have frequency dispersion and absorption. The wave shape
of the Sommerfeld precursor in the photonic crystal does not differ from that of the
Sommerfeld precursor that arises in a homogeneous medium. The instantaneous
amplitude and period of the transmitted Sommerfeld precursor in the photonic crystal
decrease with increasing spatial average of the squared plasma frequencies of the
photonic crystal slabs.
2.1 Introduction
The propagation of electromagnetic pulses in photonic crystals [1] exhibits many in-
teresting phenomena, of which the most familiar are the effects of the photonic band
gap. The photonic band gap arises as a result of Bragg-reflection of the electromag-
netic waves for certain wave-vectors and it allows photonic crystals to be applied
in for instance information technology as small-scale and low-loss waveguides, or
in fundamental research as devices that control spontaneous atomic photon emis-
sion [110]. Another effect of photonic crystals is that the magnitude of the group
velocity of electromagnetic pulses can be considerably reduced [111], if these pulses
1This chapter is based on R. Uitham and B. J. Hoenders, Opt. Comm. 262, 211-219 (2006)
20 The Sommerfeld precursor in photonic crystals
are composed of frequencies close to the edge of the photonic band gap. Theory
predicts that this group speed can approach zero in photonic crystals that have many
periods [92]. This allows for applications of photonic crystals as optical delay lines
or as data storage compounds [112]. Not only small group velocities have been ob-
served in photonic crystals, also superluminal group velocities and photon tunneling
effects have been measured [93, 113–115].
Although the applications of photonic crystals rely on the photonic band gap,
it can be expected from the theory of electromagnetic pulse propagation in homoge-
neous dielectric media with frequency dispersion and absorption [14,95,96] that there
are also interesting phenomena associated with the very high- and very low-frequency
components of an electromagnetic pulse that propagates through a photonic crystal,
that is, from the frequency components that lie outside of the photonic band gap.
When an electromagnetic pulse propagates in a homogeneous dielectric medium
with frequency dispersion and absorption, it gradually separates into distinct parts [95,
96] in configuration space. The wavefront of the pulse is composed of the infinite fre-
quency components and always propagates at the speed of light in vacuum, because
the electrons of the medium are too inert to follow the rapid oscillations of these com-
ponents of the electromagnetic field. Therefore, the infinite frequency components of
the pulse experience the homogeneous medium as if it were a vacuum. Immediately
behind the wavefront, the Sommerfeld precursor emerges and this first precursor is
composed from the very high frequency components of the pulse, which also have a
relatively weak interaction with the medium. As compared to the amplitude of the
applied optical pulse, the instantaneous amplitude of the Sommerfeld precursor is
usually very small, because the very high frequency components usually form only
a marginal part of the applied optical pulse. The amplitude and period of the Som-
merfeld precursor depend on propagation distance, time and on the squared plasma
frequency of the homogeneous medium, where the square has its origin in the fact
that, for the Lorentz medium, the plasma frequency enters the refractive index only
in squared form. Behind the first precursor there is a short period of rest after which
the Brillouin precursor emerges. This second precursor is composed from the very
low frequency components provided by the applied pulse and has, as compared to the
first forerunner, larger instantaneous amplitudes. Behind the Brillouin precursor, the
amplitude and period of the field tune to those of the applied pulse. This transition
marks the beginning of the main part of the pulse. Whereas the amplitude of the main
part of the pulse decays exponentially as a function of propagation distance, the peak
amplitudes of the precursors decay algebraically with propagation distance [95, 96].
2.2 Model for the one-dimensional photonic crystal 21
This long range persistence property of the precursors may allow for applications of
these forerunners in underwater communication or medical imaging [101].
Both precursors have been experimentally observed for microwaves transmitted
through guiding structures that have dispersion characteristics similar to those of di-
electrics [102] and for optical pulses in water and in GaAs [100, 103].
Since the precursors are strongly connected with dispersion characteristics of the
medium, it is interesting to find out how the forerunners are affected by the waveguide
dispersion that is inherently present in photonic crystals. In this chapter, the Sommer-
feld precursor theory is extrapolated from homogeneous media to one-dimensional
photonic crystals.
This chapter has been organized as follows. In Sec. 2.2 the photonic crystal
is modeled. Sec. 2.3 is devoted to the specification of the applied pulse. A brief
review of the plane wave transmission coefficient for the one-dimensional photonic
crystal is given in Sec. 2.4. In Sec. 2.5, the wavefront of the transmitted pulse is
determined and in Sec. 2.6 the Sommerfeld precursor is investigated. The influence
of the inhomogeneities of the photonic crystal medium on this precursor is discussed
in Sec. 2.7. We conclude in Sec. 2.8. In App. A, the accuracy of the calculation of
the Sommerfeld precursor is discussed.
2.2 Model for the one-dimensional photonic crystal
Our model for the stratified one-dimensional photonic crystal is the periodic multi-
layer and it has been depicted in Fig. 2.1. The multilayer is infinitely extended in
the directions perpendicular to the x-axis and the spaces to the left and the right from
the medium are vacua. Each of the layers λ = 1, . . . ,N contains two homogeneous
dielectric slabs σ = A,B. Slab σ has index of refraction nσ. The physical widths lσ of
the slabs add up to the layer width, lA + lB = l.
The frequency dependence of the slab refractive indices is obtained from the
Lorentz model for atomic polarization as
nσ(ω) =
(1+
ω2pσ
ω2σ −ω2 −2iγσω
)1/2
, (2.2.1)
where ω is the angular frequency of the electromagnetic field, ωpσ the plasma fre-
quency and γσ the damping parameter of the electron resonance of slab σ at ω = ωσ.
Fig. 2.2 depicts the frequency dependence of the slab refractive indices, for which the
22 The Sommerfeld precursor in photonic crystals
O
x
lAlB
l
nB nA · · · · · · nB nA
E0
E ′0
EA1
E ′A1
EB1
E ′B1
· · · · · ·
· · · · · ·
EBN
E ′BN
EAN
E ′AN
EN
E ′N
λ = 1, N
Figure 2.1: Model for the stratified one-dimensional photonic crystal. Each of the N
layers contains two slabs of widths lA and lB and respectively the refractive indices nA
and nB. Also shown are the amplitudes of the linearly polarized electric fields. The
arrows indicate the propagation directions of these plane waves.
parameter values are listed in the table. Also plotted in Fig. 2.2 are the expansions of
the slab refractive indices about infinite frequency up to terms quadratic in 1/ω,
nσ(ω) = 1− 1
2
ω2pσ
ω2, (2.2.2)
since these expansions will be used in the Sommerfeld precursor theory. From Fig. 2.2
it can be concluded that the multilayer is an inhomogeneous medium since values of
the slab refractive indices differ from each other. In the following section, the applied
pulse is specified.
2.3 Applied pulse 23
Re@nAD
Re@nBD
Im@nAD Im@nBD
nAï
nBï
0.5 1.0 1.5 2.0 2.5ΩΩA
-1
1
2
3
Parameters (all in units ωA)
ωpA = 1.2 ωpB = 1.5
ωA = 1.0 ωB = 1.1
γA = 0.10 γB = 0.15
Figure 2.2: Slab refractive indices as function of frequency. The dotted lines give the
quadratic expansions of the indices about infinite frequency.
2.3 Applied pulse
The linearly polarized applied electromagnetic field is perpendicularly incident from
the left on the multilayer. The amplitude of the electric components of the applied,
reflected and transmitted field are denoted respectively as E0, E ′0 and EN . In the
complex Fourier representation, the applied electric field reads as
E0(t,x) =∫
dωE0(ω;x)exp(−iωt) , (2.3.1)
where the Fourier component of the applied electric field is given by
E0(ω;x) =1
2π
∫dtE0(t,x)exp(iωt) . (2.3.2)
The Fourier component satisfies the Helmholtz equation, which reads in the vacuum
as (∂2
x + k20
)E0(ω;x) = 0, (2.3.3)
24 The Sommerfeld precursor in photonic crystals
where k0 = ω/c, with c the speed of light in vacuum. Eq. (2.3.3) has the following
rightwards propagating solution,
E0(t,x) =∫
dωE0(ω)exp(−iωt + ik0x) , (2.3.4)
where E0 (ω) = E0 (ω;x = 0) is the Fourier coefficient of the applied field at the en-
trance plane. The applied field is prepared such that, at the entrance plane, the am-
plitude of the field is nonzero only at times t ∈ [0,T ] with T positive and finite. This
gives for the electric field, that
E0(t,x = 0) = E0(t)I[0,T ](t), (2.3.5)
where I[0,T ](t) = 1 if t ∈ [0,T ] and I[0,T ](t) = 0 if t /∈ [0,T ]. Further, because the field
must be a continuous function of time, E0(0) = E0(T ) = 0. To realize this, E0(t) is
expanded in a Fourier sine series
E0(t) =∞
∑m=0
E0(ωm)sin(ωmt) , (2.3.6)
where
E0(ωm) =2
T
∫ T
0dtE0(t)sin(ωmt) , (2.3.7)
and where the carrier frequencies are given by ωm = mπ/T . With the above specifi-
cations, it can be calculated that
E0(ω) =1
2π ∑m
ωmE0(ωm)
ω2 −ω2m
((−1)m exp(iωT )−1) . (2.3.8)
Only applied pulses with finite carrier frequencies will be considered, so there exists
a nonnegative integer M such that
E0(ωm) = 0 for m > M. (2.3.9)
Eqs. (2.3.4) and (2.3.8) together describe a perpendicularly incident plane wave elec-
tric pulse of finite time duration. The transmitted field that results from this applied
plane wave packet is obtained via the plane wave transmission coefficient for the
multilayer, which is calculated in the next section.
2.4 Plane-wave transmission coefficient for the multilayer 25
2.4 Plane-wave transmission coefficient for the multilayer
The amplitude of the rightwards propagating electric field in slab σ of layer λ is
denoted as Eσλ and the amplitude of the leftwards propagating electric field in the
same slab of the same layer is indicated with a prime as E ′σλ, see Fig. 2.1. The electric
fields in slabs A of two successive layers λ−1 and λ in the multilayer of Fig. 2.1 are
related via the corresponding unimodular single-layer transfer matrix [116],
TA =
(A1 B1
C1 D1
), (2.4.1)
as (EAλ
E ′Aλ
)= TA
(EAλ−1
E ′Aλ−1
). (2.4.2)
The single-layer transfer matrix is constructed from respectively the propagation and
dynamical matrices,
Pσ = diag(exp(ikσlσ) ,exp(−ikσlσ)) , (2.4.3)
∆σ =
(1 1
−nσ/(µ0c) nσ/(µ0c)
), (2.4.4)
where kσ = k0nσ and where the dynamical matrices are given for perpendicularly
incident fields, as
TA = PA∆−1A ∆BPB∆−1
B ∆A. (2.4.5)
The entries of TA follow from Eq. (2.4.5) as
A1 = exp(ikAlA)
(cos(kBlB)+
i
2(nB/nA +nA/nB)sin(kBlB)
),
B1 =i
2exp(−ikAlA)(nB/nA −nA/nB)sin(kBlB) ,
C1 =−i
2exp(ikAlA)(nB/nA −nA/nB)sin(kBlB) ,
D1 = exp(−ikAlA)
(cos(kBlB)− i
2(nB/nA +nA/nB)sin(kBlB)
).
(2.4.6)
If, instead of having the multilayer surrounded by vacuum, a dielectric medium with
index of refraction nA would surround the multilayer, then the electric fields at both
26 The Sommerfeld precursor in photonic crystals
sides from the multilayer are related by a product of N matrices TA. The entries of
T NA ≡
(AN BN
CN DN
), (2.4.7)
are found by solving Eq. (2.4.2) with the discrete Mellin transform method [117].
One finds
AN = A1
αN −α−N
α−α−1− αN−1 −α−N+1
α−α−1,
BN = B1
αN −α−N
α−α−1,
CN = C1
αN −α−N
α−α−1,
DN = D1αN −α−N
α−α−1− αN−1 −α−N+1
α−α−1.
(2.4.8)
Here α±1 = 12trTA ±
√(12trTA
)2 −1 are the eigenvalues of TA. The correction of the
surrounding medium A to a vacuum gives that the amplitudes on both sides of the
multilayer are related as
(EN
E ′N
)= ∆−1
0 ∆AT NA ∆−1
A ∆0
(E0
E ′0
), (2.4.9)
where
∆0 =
(1 1
−1/(µ0c) 1/(µ0c)
). (2.4.10)
The transmission coefficient of the multilayer surrounded by the vacuum follows as
tN ≡(
EN
E0
)∣∣∣E ′
N=0=
−4nA
(nA −1)2AN − (n2A −1)(CN −BN)− (nA +1)2DN
. (2.4.11)
Via the transmission of the applied pulse through the multilayer, one arrives at the
transmitted pulse. In the following section, the speed of propagation of the wavefront
of the transmitted pulse will be determined.
2.5 Wavefront of the transmitted pulse
In order to describe the Sommerfeld precursor of the transmitted pulse, which fol-
lows immediately behind the wavefront of the pulse, the wavefront itself will first be
2.5 Wavefront of the transmitted pulse 27
determined. Hereto, consider the expression for the transmitted pulse, evaluated at
the exit plane at x = Nl,
EN(t,x = Nl) =∫
dωtNE0 exp(−iωt) . (2.5.1)
A pulse of finite time duration necessarily contains components with infinite absolute
frequency. This can be seen from considering the integrand of Eq. (2.5.1) under the
limit to absolute infinite frequency in different directions in the complex frequency
plane. In the following, the contributions of these components will be investigated.
Causality implies that the integrand of Eq. (2.5.1) is analytic at and above the real
frequency axis [118]. Hence the integration path may freely be deformed from the
real axis to path S, which has been illustrated in Fig. 2.3. The path S coincides with
the real frequency axis up to a semicircle detour in the upper half of the complex
frequency plane with center at the origin of the frequency plane and radius Ω. The
Re [ω]
Im [ω]
0
Ω
S
b b b b b b b bb b b
−ωM ωM−ωm
γσ,ωσ,ωpσ
ωm
Figure 2.3: Illustration of the integration path S, which follows the real frequency axis
up to a semicircle detour with center at the origin of the complex frequency plane and
radius Ω. The radius is chosen much larger than the various frequency parameters of
the multilayer and the applied pulse, which have been indicated on the real axis.
indices of refraction satisfy
lim|ω|→∞
nσ = 1, (2.5.2)
28 The Sommerfeld precursor in photonic crystals
and the multilayer transmission coefficient satisfies
tN |nσ=1 = exp(ik0Nl) . (2.5.3)
Therefore, within the zeroth order expansion of the slab refractive indices about infi-
nite frequency, the transmitted electric field reads as
EN(t,x = Nl)|nσ=1 =∫
SdωE0 exp(−iω(t −Nl/c)) . (2.5.4)
If the spacetime coordinate
τ ≡ t −Nl/c (2.5.5)
takes on negative values, the integrand in Eq. (2.5.4) vanishes far above the real
frequency axis. The contribution from the part of the integration near and at the
real axis decreases as ω−2 far from the origin as a result of the factor E0. Therefore,
for τ < 0 the amplitude of the transmitted field is equal to zero. For τ > 0, exponential
decay of (part of) the integrand is realized with an integration path that is deformed far
away into the lower half frequency plane. But with such a deformation, the poles from
the slab refractive indices of Eq. (2.2.1) at ω =±√
ω2σ − γ2
σ + iγσ and the poles of the
Fourier coefficient of the applied field of Eq. (2.3.8) at ω = ±ωm are encountered.
This results in a nonzero signal for τ > 0, hence the arrival of the wavefront at the
exit plane at x = Nl is given by the equation
τ = 0. (2.5.6)
The quantity τ(t,x) is therefore the time elapse after the wavefront has passed the exit
plane. Now that the wavefront of the electromagnetic pulse in the photonic crystal
has been determined, the transmitted pulse immediately behind the wavefront can be
investigated.
2.6 Sommerfeld precursor
The Sommerfeld precursor starts immediately behind the wavefront. In terms of the
coordinate τ, the transmitted field of Eq. (2.5.1) reads as
EN(τ,x = Nl) =∫
SdωtN exp(−iωτ− ik0Nl) E0. (2.6.1)
Here, S is the integration path of Fig. 2.3. The wavefront was determined by approxi-
mating the refractive indices by their values at infinite frequency. For the Sommerfeld
2.6 Sommerfeld precursor 29
precursor, the indices of refraction are expanded about infinite frequency up to and
including terms quadratic in 1/ω, these expansions are given by nσ in Eq. (2.2.2).
To determine tN |nσ=nσ , it is instructive to consider another expanded form for the
transmission coefficient. The Fresnel reflection coefficient for an electric field that
propagates in slab σ and is reflected at a boundary with slab σ′ and the corresponding
Fresnel transmission coefficient are given, for perpendicular incident fields, respec-
tively by [106]
rσσ′ =nσ −n′σnσ +n′σ
,
tσσ′ = 1+ rσσ′ .
(2.6.2)
Under horizontal propagation through a slab σ, the field acquires a ’slab-propagation
factor’ given by
pσ = exp(ikσlσ) . (2.6.3)
The single-layer transfer matrix elements of Eq. (2.4.6) are, expressed in Fresnel
coefficients and slab-propagation factors,
A1 = tABtBA pA(pB − r2AB p−1
B ),
B1 = rBAt−1AB t−1
BA pA(pB − p−1B ),
C1 = rABt−1AB t−1
BA p−1A (pB − p−1
B ),
D1 = t−1AB t−1
BA p−1A (p−1
B − r2AB pB).
(2.6.4)
Now we expand the transmission coefficient tN in powers of pA and pB. This expan-
sion is up to and including terms at order pN+2A and pN+2
B equal to
tN = t0BtN−1AB tN
BA pNA pN
B tA0
(1+ rBArB0 p2
B +(N −1)r2AB p2
A
+(N −1)r2BA p2
B + rA0rAB p2A
), (2.6.5)
where the Fresnel coefficients that bear a subscript zero are used for the interfaces of
the multilayer with the surrounding vacuum. The paths corresponding to the terms
in Eq. (2.6.5) have been sketched in Fig. 2.4. With Eq. (2.2.2), the high-frequency
expansions of the Fresnel coefficients of Eq. (2.6.2) up to terms quadratic in 1/ω
follow as
rσσ′(ω) =1
4
ω2pσ′ −ω2
pσ
ω2,
tσσ′(ω) = 1+1
4
ω2pσ′ −ω2
pσ
ω2.
(2.6.6)
30 The Sommerfeld precursor in photonic crystals
nB nA nB nA · · · · · · nB nA
λ = 1, 2, N
t0BtN−1AB tN
BAtA0 pNA pN
B
t0BrABrB0tN−1AB tN
BAtA0 pNA pN+2
B
t0BtN−1AB tN
BAtA0r2BA pN
A pN+2B
t0BtN−1AB tN
BAtA0r2AB pN+2
A pNB
t0BtN−1AB tN
BArA0rABtA0 pN+2A pN
B
11
N −1
N −1
1
Figure 2.4: First few terms in the path-length ordered form of the transmission coef-
ficient.
Therefore, within the high-frequency expansion, the transmitted light-rays that have
experienced reflections inside the multilayer give contributions of fourth and higher
orders in 1/ω, so only the straight light-ray survives the high-frequency expansion.
When the terms that result from the expanded indices of refraction are kept up to and
including second order in 1/ω, one thus finds
tN |nσ=nσ = exp
(ik0Nl
(1− 1
2
ω2p
ω2
)), (2.6.7)
where
ω2p ≡
lAω2pA + lBω2
pB
l(2.6.8)
is the spatial average of the squared slab plasma frequencies. The expansion of the
Fourier coefficient of the applied pulse, Eq. (2.3.8), about infinite frequency up to
2.6 Sommerfeld precursor 31
and including quadratic terms in 1/ω is given by
E0(ω)||ω|>>ωM= − 1
2π
1
ω2 ∑m
ωmE0(ωm), (2.6.9)
so that the high-frequency contribution to the transmitted field, Eq. (2.6.1), follows
as
EN(τ,x = Nl) = − 1
2π ∑m
ωmE0(ωm)∫
Sdω
exp(−iωτ− iξ/ω)
ω2, (2.6.10)
where
ξ =Nl
2cω2
p. (2.6.11)
The steps taken in the following paragraph, in order to perform the integration in
Eq. (2.6.10), are merely a repetition of the work of Brillouin [96].
Consider the path S, which is obtained from path S by reflection about the point
ω = 0. On S, Eq. (2.6.10) vanishes for infinitely large Ω if τ > 0. Hence, when,
for τ > 0, this path S is added to S in Eq. (2.6.10), zero is added to the integral.
The integration over S is chosen to run from ω = +∞ to ω = −∞. When S is added
to S, one obtains a circular path, denoted as C, which is traversed clockwise. With
reversion to counterclockwise traversing of C, and with rewriting the exponent of
Eq. (2.6.10), one obtains
EN(τ,x = Nl) =1
2π ∑m
ωmE0(ωm)∮
Cdω
exp
(−i√
ξτ
(ω√
τξ+ 1
ω
√ξτ
))
ω2. (2.6.12)
Define the new integration variable φ by
exp(iφ) =
√τ
ξω, (2.6.13)
so that dω/ω2 = i√
τ/ξexp(−iφ)dφ. Integration over φ from zero to 2π corre-
sponds to integration along the contour C with radius Ω =√
ξ/τ in the complex
plane. Therefore, for small τ, the integration path lies far away from the origin of the
frequency plane and the initially transmitted electric field can be identified as
EN(τ,x = Nl) = ∑m
ωmE0(ωm)
√τ
ξJ1
(2√
ξτ)
. (2.6.14)
32 The Sommerfeld precursor in photonic crystals
Here J1 is the Bessel function of the first kind and first order. Eq. (2.6.14) has the
same form as the expression for the Sommerfeld precursor for propagation in a homo-
geneous medium [96]. For every photonic crystal of the form treated in this chapter,
there exists an equivalent homogeneous medium with plasma frequency ωp, defined
by Eq. (2.6.8), such that both media give rise to the same Sommerfeld precursor for
the applied pulses of the form treated in this chapter. The amplitude and period of
the Sommerfeld precursor in the photonic crystal depend, through ξ, on the spatial
average of the squared slab plasma frequencies. In the next section, this dependence
will be discussed.
2.7 Discussion
Fig. 2.5 shows the field of Eq. (2.6.14) as a function of time τ for transmission through
a multilayer with N = 100 and l = 600nm. The plots are given for three different
values of ω2p, which are expressed in units of the squared plasma frequency of silicon
[2, p. 278], ω2pSi =
(2.4 ·1016
)2s−2. From Fig. 2.5, it follows that the amplitude and
period of the Sommerfeld precursor decrease with increasing ω2p.
At last, the initial amplitude of the transmitted Sommerfeld precursor, Eq. (2.6.14),
is compared to the amplitude of the applied signal, Eq. (2.3.6). This comparison is
done for pulse transmission through the multilayer that has ωp = ωpSi, for which
the Sommerfeld precursor is given by the solid line in Fig. 2.5. For ωp = ωpSi,
Eq. (2.6.11) gives ξ = 5.76 · 1019s−1. The first maximum of J1 is at 2√
ξτ = 1.84
and has the value 0.582. At this maximum, τ = 1.47 ·10−20s. For simplicity, we take
an applied pulse with only one single carrier frequency ωc and amplitude E0(ωc), so
that
E0(t,x = 0) = I[0,T ](t)E0(ωc)sinωct. (2.7.1)
For an optical carrier frequency ωc = 3.0 ·1015s−1, At the first maximum of the Bessel
function, Eq. (2.6.14) gives EN = 2.8 · 10−5E0(ωc). Therefore, the initial amplitude
of the transmitted Sommerfeld precursor is very small compared to the amplitude of
the applied pulse.
2.8 Conclusion
The wavefront of an electromagnetic plane wave pulse that propagates in a dielectric
medium is constructed from the infinite frequency components of that pulse. The
2.8 Conclusion 33
Ωp2
ΩpSi2=1
Ωp2
ΩpSi2=1.2
Ωp2
ΩpSi2=0.8
2 4 6 8 10Τ in 10
-19s
-0.0010
-0.0005
0.0005
0.0010
multilayer dimensions
N = 100 l = 600nmEN in units ∑mωm
ωpSiE0(ωm)
Figure 2.5: Sommerfeld precursor field as a function of time, for transmission
through the multilayer specified in the table and with three different spatially aver-
aged squared slab plasma frequencies, given in units of the squared plasma frequency
of silicon, ω2pSi =
(2.4 ·1016
)2s−2.
physical explanation for this is that the electrons of the medium are too inert to fol-
low the infinitely fast oscillations of these components of the electric field. As a
consequence, the medium is not polarized by these components. Therefore, these
components experience the medium as if it were a vacuum. When a homogeneous
dielectric medium is replaced by a one-dimensional photonic crystal that consists of
layers of dielectric media, the same holds and the wavefront still propagates at the
vacuum speed of light.
The Sommerfeld precursor results from very high frequency components of the
pulse, which experience the medium as if it were almost vacuum. This precursor
immediately follows the wavefront of the transmitted pulse. Although the very high-
34 The Sommerfeld precursor in photonic crystals
frequency components applied pulse do experience reflection against the interfaces
of the periodic multilayer, because there is a contrast in the refractive indices of the
slabs for these frequencies, the transmitted Sommerfeld precursor still has the same
shape as it does for propagation in a homogeneous medium. The only difference is
that in the expression for the Sommerfeld precursor, the squared plasma frequency for
the case of a homogeneous medium is replaced by the spatial average of the squared
plasma frequencies of the slabs of the multilayer. The amplitude and period of the
Sommerfeld precursor decrease when this spatial average of the squared slab plasma
frequencies increases.
Chapter 3
The Brillouin precursor in
photonic crystals
In this chapter1, we calculate the Brillouin precursor of an electromagnetic pulse
that has been transmitted through a stratified one-dimensional photonic crystal with
frequency dispersion and absorption. The slab contrast of the photonic crystal affects
the precursor field after a certain rise time. Then, the frequency spectrum of the
Brillouin precursor starts to peak at the scattering resonances of the medium whereas
minima appear at the Bragg-scattering frequencies.
3.1 Introduction
Photonic crystals [1] have recently gained much interest, because the propagation
of light can be efficiently controlled with these materials. A photonic crystal is a
spatially repeated structure, or unit cell, of various dielectric components that each
individually have in general a different interaction strength with an electromagnetic
field so that an incident electromagnetic field is reflected periodically. Exactly as
in the case of electrons in interaction with a periodic atomic lattice, where a band
gap appears in the electron dispersion relation due to Bragg scattering, the lattice
of material components in photonic crystals creates a band gap for electromagnetic
radiation. For the frequencies inside this gap, no propagating wave solutions exist
1This chapter is based on a paper of R. Uitham and B. J. Hoenders that has been accepted for
publication in Opt. Comm.
36 The Brillouin precursor in photonic crystals
inside the crystal. Another interesting effect of photonic crystals is that the group
velocity of an electromagnetic pulse can be reduced considerably [92, 93] when the
pulse is predominantly composed of frequencies that lie close to the edge of the band
gap [111]. Therefore, photonic crystals open new avenues to manipulate the prop-
agation of an electromagnetic field. The expected applications of photonic crystals
are numerous, for instance waveguides [1], diodes [119], data storage compounds
and delay lines [92], lasers [120] and devices that control the spontaneous atomic
emission of photons [110]. However, it is still difficult to grow highly regular three-
dimensional structures with lattice constants of only a few hundreds of nanometers.
The large-scale fabrication of three-dimensional photonic crystals with photonic band
gaps in the visible part of the frequency spectrum is therefore still limited.
As the fabrication of photonic crystals is developing, various aspects of the theory
of electromagnetic pulse propagation in homogeneous media can be reformulated in
order to apply to photonic crystals. In 1914, Sommerfeld and Brillouin calculated that
two electromagnetic precursors arise when an electromagnetic pulse propagates in a
linear, isotropic, homogeneous dielectric material with frequency dispersion and ab-
sorption modeled as a single-electron resonance Lorentz medium [94]. These precur-
sors are small-amplitude and high-frequency field oscillations that propagate ahead
of the main part of the pulse. The forerunners have been experimentally observed
for the first time in 1969 by Pleshko and Palocz [102]. The fastest propagating one,
the Sommerfeld precursor, is composed from the very high-frequency components
of the applied pulse, where very high means as compared to the electron resonance
frequency of the medium. For these high frequencies, the response of the medium
to the field very much resembles that of a vacuum, which explains the fast propaga-
tion and slow decay of this first precursor. After the Sommerfeld precursor follows
the Brillouin precursor, which is composed from the very low-frequency components
of the applied pulse, where low frequency again means as compared to the electron
resonance frequency. These components interact relatively weak with the medium as
well. After the Brillouin precursor, the main part of the applied pulse follows.
Because the precursors are strongly tied up with the dispersion characteristics of
the medium, it is to be expected that the forerunners that arise in a photonic crys-
tal differ from those that arise in a homogeneous medium. In the previous chapter,
the Sommerfeld precursor was calculated for electromagnetic pulse transmission in
a one-dimensional photonic crystal. For that calculation, the indices of refraction of
the photonic crystal slabs were expanded about infinite frequency. Within this ex-
pansion, the Fresnel reflection coefficients that belong to interfaces of the photonic
3.2 Model for the photonic crystal 37
crystal vanish up to and including the second order terms so that only the transmit-
ted light-ray that has not been reflected between the interfaces of the photonic crys-
tal contributes to the Sommerfeld precursor. As a result, the Sommerfeld precursor
merely feels the spatial average of the medium, no effects from the inhomogeneity of
the medium on the Sommerfeld precursor were found. From this point of view, it is
to be expected that the Brillouin precursor will be influenced stronger by the medium
inhomogeneities since the slab contrast does not vanish at low frequencies. In this
chapter, the Brillouin precursor theory is extrapolated from homogeneous media to
one-dimensional photonic crystals.
This chapter has been organized as follows. In Sect. 3.2, the photonic crystal is
modeled and its transmission coefficient for the electromagnetic field is derived in
Sect. 3.3. The transmittance of the medium is analyzed numerically in Sect. 3.4.
Thereafter, in Sect. 3.5 we discuss how to apply the method of steepest descent
in order to calculate the Brillouin precursor. Then, in Sect. 3.6, we calculate the
transmitted Brillouin precursor resulting from a delta-peak input pulse and from a
step-modulated sinusoidal input field. In Sect. 3.7 the results are discussed. Finally,
conclusions are drawn in Sect. 3.8.
3.2 Model for the photonic crystal
Our model for the one-dimensional photonic crystal is a periodic multilayer which
has been depicted in Fig. 3.1. The x-axis is taken as the principal axis of the crystal.
The crystal consists of N layers of physical width l and each layer contains two
homogeneous slabs, denoted as slab A and slab B, respectively of physical widths lAand lB that add up to the layer width, lA + lB = l. The coordinates of the interfaces of
the multilayer are given as
xmn = xL +(n−1) l +δmBlA, m = A,B, n = 1, . . . ,N,xR = xL +Nl.
(3.2.1)
The interfaces at x = xL and at x = xR are respectively referred to as the entrance and
exit interface. To the left and to the right from the multilayer, there are respectively
the homogeneous materials L and R. All homogeneous media m = A,B,L,R give an
isotropic and linear response to the electromagnetic field so that these materials are
fully characterized with the scalar permittivities and permeabilities. The responses
38 The Brillouin precursor in photonic crystals
L A A AB B B R
x
lA lB
l
εL εA εB · · · · · · εA εB εR
µL µA µB · · · · · · µA µB µR
xA1 xB1 xA2 xAN xBN xR
EA1
HA1
EB1
HB1
· · · · · ·· · · · · ·
EAN
HAN
EBN
HBN
EL
HL
ER
HR
Figure 3.1: Model for the stratified one-dimensional photonic crystal. Slabs A and B
respectively have physical widths lA and lB and together form a layer of thickness l.
The interface coordinates are given along the x-axis. The permittivities and perme-
abilities of material m = L,A,B,R are respectively given by εm and µm. Also indicated
are the electric (E) and magnetic (H) fields in the various homogeneous subspaces
of the system
of the media m to the electric field are modeled as Lorentz media with single2 elec-
tron resonances [94], whereas there is no interaction with the magnetic field. The
absolute permittivity and absolute permeability of medium m are therefore given by
respectively
εm = ε0 +ε0ω2
pm
ω2m −2iγmω−ω2
, (3.2.2a)
µm = µ0, (3.2.2b)
where ω is the angular frequency of the electromagnetic field, ε0 and µ0 respectively
the vacuum permittivity and permeability, ωm the electron resonance frequency, ωpm
the plasma frequency and γm the absorption parameter of medium m. Now that the
2For the evolution of precursors in homogeneous media with multiple electron resonances, see
Ref. [97].
3.3 Transmission coefficient of the photonic crystal 39
photonic crystal has been modeled, its transmission coefficient for the amplitude of
the electric component of an electromagnetic field will be calculated in the following
section.
3.3 Transmission coefficient of the photonic crystal
For simplicity, the applied field is incident perpendicularly on the multilayer. The
theory allows for an extension to oblique incidence with separation in TE- and TM-
polarization, but since the interest is only in the effect of the medium inhomogeneities,
this extension would merely obscure the purpose of the investigation. The system
of the multilayer plus the two surrounding media consists of 2N + 2 homogeneous
subspaces. The homogeneous subspaces are labeled as mn, where the first index
m = A,B,L,R indicates the material and the last index n = 1, . . . ,N indicates the layer
number. The latter index is present only if m = A,B, see Fig. 3.1. The real amplitude
of the electric component of the linearly polarized electromagnetic field in subspace
mn reads in a Fourier representation as
Emn(t,x) =∫
dω Emn(ω;x)exp(−iωt) . (3.3.1)
where the complex Fourier coefficient is given by
Emn(ω;x) =1
2π
∫dt Emn(t,x)exp(iωt) . (3.3.2)
These coefficients obey Helmholtz’ equation,
(∂2
x + k2m
)Emn(ω;x) = 0, (3.3.3)
where
km = ω√
εmµm. (3.3.4)
The solutions to Eq. (3.3.3) give, after substitution into Eq. (3.3.1), that the electric
field consists of respectively the right- and leftwards propagating parts
E(r)mn(t,x) =
∫dω E
(r)mn(ω)exp(−iωt + ikm (x− xmn)) , (3.3.5a)
E(l)mn(t,x) =
∫dω E
(l)mn(ω)exp(−iωt − ikm (x− xmn)) , (3.3.5b)
40 The Brillouin precursor in photonic crystals
where the Fourier coefficients of respectively the right- and leftwards propagating
parts of the electric field, as evaluated at the interfaces, are given by
E(r)mn(ω) =
1
2π
∫d tE
(r)mn(t,xmn)exp(iωt) , (3.3.6a)
E(l)mn(ω) =
1
2π
∫dt E
(l)mn(t,xmn)exp(iωt) . (3.3.6b)
From Maxwell’s equation ∇× E− iωB = 0 with B = µH, the Fourier coefficients
of respectively the right- and leftwards propagating parts of the magnetic field that
correspond to those of the electric fields of Eqs. (3.3.6) follow as
H(r)mn = −E
(r)mn/Zm, (3.3.7a)
H(l)mn = E
(l)mn/Zm, (3.3.7b)
where Zm =√
µm/εm is the impedance of material m. The left-to-right transmission
coefficient of the photonic crystal for the electric field amplitude is defined as
tN ≡(
E(r)R /E
(r)L
)∣∣∣E
(l)R =0
, (3.3.8)
and in the rest of this section we will calculate tN . Let xm′n′ denote the coordinate
of the interface immediately to the right of the interface at x = xmn. The tangential
components of the electric and magnetic fields must be continuous at each interface,
hence
Emn (t,xm′n′) = Em′n′ (t,xm′n′) , (3.3.9a)
Hmn (t,xm′n′) = Hm′n′ (t,xm′n′) . (3.3.9b)
With Eqs. (3.3.1), (3.3.6) and (3.3.7), Eqs. (3.3.9) give
(E
(r)m′n′
E(l)m′n′
)= Dm′mPm
(E
(r)mn
E(l)mn
), (3.3.10)
where the transmission matrix Dm′m is constructed from the dynamical matrices
∆m =
(1 1
−Z−1m Z−1
m
)(3.3.11)
3.3 Transmission coefficient of the photonic crystal 41
as
Dm′m = ∆−1m′ ∆m. (3.3.12)
Note that by construction D−1m′m = Dmm′ and Dm′mDmm′′ = Dm′m′′ . Further, in Eq. (3.3.10),
the unimodular propagation matrices are given by
Pm = diag(exp(ikmlm) ,exp(−ikmlm)) . (3.3.13)
Eq. (3.3.10) relates the Fourier coefficients of the left- and rightwards propagating
electric field components in subsequent slabs. The single-layer transfer matrix in-
volves transfer over slabs A and B and is given by
TA = DABPBDBAPA. (3.3.14)
The entries of TA =
(A1 B1
C1 D1
)can readily be calculated as
A1 = exp(ikAlA)
(coskBlB +
i
2
(ZA
ZB
+ZB
ZA
)sinkBlB
), (3.3.15a)
B1 =i
2exp(−ikAlA)
(ZA
ZB
− ZB
ZA
)sinkBlB, (3.3.15b)
C1 =−i
2exp(ikAlA)
(ZA
ZB
− ZB
ZA
)sinkBlB, (3.3.15c)
D1 = exp(−ikAlA)
(coskBlB −
i
2
(ZA
ZB
+ZB
ZA
)sinkBlB
). (3.3.15d)
With a slight simplification by using the aforementioned properties of Dmn, the left-
to-right transfer matrix of the multilayer, T , can be constructed from the propagation
and transmission matrices as
T = DRAT NA DAL. (3.3.16)
Note that detT = ZR/ZL. The unimodularity of TA implies that the entries of the
transfer matrix for N layers,
T NA ≡
(AN BN
CN DN
), (3.3.17)
42 The Brillouin precursor in photonic crystals
are related to those of the single-layer transfer matrix as [116]
AN = A1UN−1(T1)−UN−2(T1), (3.3.18a)
BN = B1UN−1(T1), (3.3.18b)
CN = C1UN−1(T1), (3.3.18c)
DN = D1UN−1(T1)−UN−2(T1), (3.3.18d)
where the Um are the Chebyshev U-polynomials,
Um(T1) =⌊m
2⌋
∑n=0
(−1)n
(m−n
n
)(2T1)
m−2n, (3.3.19)
having as argument T1 ≡ 12trTA. In Eq. (3.3.19) the upper limit to the sum, ⌊m
2⌋,
denotes the floor function of m/2, which gives the largest integer that is smaller than
or equal to m/2. From the defining equation of the transfer matrix,
(E
(r)R
E(l)R
)=
(T11 T12
T21 T22
)(E
(r)L
E(l)L
), (3.3.20)
the left-to-right3 transmission coefficient of the multilayer for the electric field fol-
lows from Eq. (3.3.8) in terms of the T -matrix entries as
tN =detT
T22
. (3.3.21)
The left-to-right transmitted electric field amplitude, evaluated at the exit plane of the
multilayer, follows from Eqs. (3.3.5a) and (3.3.8) as
E(r)R (t) =
∫dωtNE
(r)L exp(−iωt) . (3.3.22)
In the rest of this chapter, the superscript (r) on the transmitted field will be omitted.
In the following section, the multilayer transmittance will be considered for complex
frequencies in order to determine, as a function of time, the dominant contributions to
the transmitted field. This will be done for early times, so that the initial transmitted
field is obtained.
3The right-to-left transmission coefficient equals t ′N ≡(
E(l)L /E
(l)R
)∣∣∣E
(r)L =0
= T−122 , the two transmis-
sion coefficients are therefore related as ZRt ′N = ZLtN .
3.4 Transmittance of the photonic crystal 43
3.4 Transmittance of the photonic crystal
For the application of the method of steepest descent to Eq. (3.3.22), the transmis-
sion coefficient is written as tN = exp(ln tN) and the natural time coordinate for the
multilayer is used, θ = ct/(Nl). With these substitutions, Eq. (3.3.22) becomes
ER (θ) =∫
dω EL (ω)expΦN (θ;ω) , (3.4.1)
where the complex phase function is given by
ΦN = ln tN − iω(Nl/c)θ. (3.4.2)
The transmittance TN of the multilayer is the ratio of the intensity of the transmitted
field to that of the applied field. From Eqs. (3.4.1) and (3.4.2), the transmittance
follows as TN = exp(2ReΦN). The function
XN (ξ,η;θ) = ReΦN (ω;θ) , (3.4.3)
where ξ = Reω and η = Imω, gives a half times the logarithm of the transmittance
for complex frequencies as a function of time. From plotting XN at successive times,
one can determine the dominant frequency contributions to the transmitted field as
time proceeds. Fig. 3.2 shows various plots of XN at successive instants of time, for
a photonic crystal with N = 1. The slab parameters that were used for Fig. 3.2 have
been listed in Table 3.4.1.
The value of XN is constant on contour lines andSlab parameter values
lA = 20nm lB = 30nm
ωA = 1.6 ωB = 1
ωpA = 1.8 ωpB = 0.8γA = 0.08 γB = 0.06
Table 3.4.1: Slab parameter
values, where frequency pa-
rameters are in units of ωB.
this value has been indicated for two neighboring con-
tour lines in each plot. The difference of the values of
XN on neighboring contour lines is the same for every
pair of neighboring contour lines. The plots cover pos-
itive values of ξ only, because XN is symmetric about
the imaginary axis because the field must be real. The
dominant stationary points of ΦN are those stationary
points at which XN takes on the largest values. The
three dominant stationary points of ΦN have been in-
dicated in Fig. 3.2 as φI , φII and φIII . The stationary points of ΦN are saddle-points
of XN , see App. B. At the plotting times, all stationary points in Fig. 3.2 are of first
order, as can be concluded from the variation of XN in the neighborhood of these
points, see App. B.
44 The Brillouin precursor in photonic crystals
XN at Θ=1.3
0
-0.05
PolePole
Pole
ΦI
ΦII ΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
BXN at Θ=1.4
0-0.05
PolePole
Pole
ΦI
ΦIIΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
XN at Θ=1.5
0
0.05
PolePole
Pole
ΦI
ΦIIΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
XN at Θ=1.8
00.05
PolePole
Pole
ΦI ΦII ΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
XN at Θ=2.0
00.08
PolePole
Pole
ΦIΦII
ΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
XN at Θ=2.5
00.10
PolePole
Pole
ΦI
ΦII
ΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
Figure 3.2: Contourplots of XN for N = 1 and the parameters given in Table 3.4.1 at
increasing values of time θ. The integration path (dashed curve) passes the dominant
stationary points of the phase function along the steepest descent lines of XN .
3.5 Investigation of Brillouin precursor with steepest descent method 45
3.5 Investigation of Brillouin precursor with steepest de-
scent method
According to the method of steepest descent, the instantaneous integration path,
which has been indicated in Fig. 3.2 with the dashed curve, passes through the dom-
inant stationary points. Away from the stationary points, the integration path follows
the lines of steepest descent of XN . Within the approximation of the method of steep-
est descent, the contribution to the electric field of Eq. (3.4.1) that comes from the
integration in the neighborhood of a first-order stationary point of the phase function
that follows the trajectory ω = φs (θ) is given by (see App. B)
E(s)R (θ) = EL (φs)Π(θ,φs) , (3.5.1)
where
Π(θ;φs) =√
2π∣∣∣Φ(2)
N (φs)∣∣∣− 1
2exp(ΦN (θ;φs)+ iαN (φs)) , (3.5.2)
with Φ(n)N = dnΦN/dωn and where αN (φs) = (π/2)− (1/2)arg Φ
(2)N (φs) is the angle
of the steepest descent line of XN with the ξ-axis at the stationary point. Eq. (3.5.1)
holds along the complete trajectory if the phase function is stationary to first order
along the trajectory. In order to verify this, the function
UN = ReΦ(1)N (3.5.3)
has been plotted in Fig. 3.3, again for N = 1 and for the slab parameters of Table 3.4.1.
In Fig. 3.3, the roots of the second-order derivative of the phase function are visible
as saddle-points of UN . Since UN is independent of time, the positions of the roots
of the second-order derivative of the phase function are fixed. The roots of first-
and second-order derivative of the phase function, as determined from respectively
Figs. 3.2 and 3.3, have been plotted together in Fig. 3.4. In this figure, the dashed
lines connect successive instantaneous stationary points of the phase function and
thus roughly represent the observed trajectories. Fig. 3.4 shows that the roots of
first- and second-order derivatives of the phase function do not coincide, hence the
stationary points are always of first order at the plotted time interval. Therefore, it is
allowed to use Eq. (3.5.1) for a calculation of the transmitted field.
The expression for the transmitted field of Eq. (3.5.1) contains the stationary point
trajectory ω = φs (θ). The algebraic expression for this trajectory is implicit in the
stationary phase equation
Φ(1)N = 0, (3.5.4)
46 The Brillouin precursor in photonic crystals
UN
PolePole
Pole
FNH2L=0
FNH2L=0
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
Figure 3.3: Plot of UN = ReΦ(1)N . The roots of Φ
(2)N are visible as saddle-points of UN .
and it is obtained by solving this relation for ω in terms of θ. When the instantaneous
location of a stationary point of the phase function is determined from the transmit-
tance plots as ω = ωs at some observation time θ = θs, its trajectory at times close to
this evaluation time follows from Eq. (3.5.4) together with φs (θs) = ωs and is given
in a series expansion by [121, 122]
φs (θ) = ωs +∞
∑l=1
(−1)l
l!
[∂l−1
ω
(ΦN
(θ; ω
)− ω
)l]∣∣∣∣
(θ,ω)=(θ−θs,0)
, (3.5.5)
where the auxiliary function ΦN
(θ; ω
)= Φ
(1)N
(θ+θs; ω+ωs
)/Φ
(2)N (θs;ωs) is well-
defined if the observed stationary point is of first order. For the time interval that is
spanned by the plots of Fig. 3.2, it has already been verified above that this require-
ment is fulfilled. The trajectories, as calculated from Eq. (3.5.5) with observation
time θs = 1.8, are shown in Fig. 3.4 as solid lines. These do diverge a little from
the observed trajectories at times that differ much from the observation time. This
divergence stems from the fact that the sum in Eq. (3.5.5) could only be taken up to
and including the third term because of a shortage of computer power.
3.5 Investigation of Brillouin precursor with steepest descent method 47
1.3
1.3
1.3
1.4
1.4
1.4
1.4
1.4
1.4
1.5
1.5
1.5
1.5
1.5
1.51.8
1.8 1.8
1.8
1.8 1.8
2.0
2.02.0
2.0
2.02.0
2.5
2.5
2.5
FNH1L=0
FNH1L=0
FNH1L=0
FNH2L=0
FNH2L=0
à
à
à
Pole Pole
Pole
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
Roots of FNH1L
and FNH2L
Figure 3.4: Roots of Φ(1)N and Φ
(2)N as obtained from respectively the contourplots of
XN and UN . The dashed lines connect the roots of Φ(1)N at the indicated successive
values of time θ. The solid lines represent the stationary point trajectories ω = φI (θ)(leftmost solid line), φII (θ) (middle) and φIII (θ) (right) in the time interval 1.3 ≤ θ ≤2.5, as calculated from the series expansion formula about evaluation time θs = 1.8.
The symmetry Φ∗N (ω) = ΦN (−ω∗), which follows from reality of the field, im-
plies that, if the phase function has a stationary point at ω = φs, it also has one at
ω = −φ⋆s . When an integration path is used that is symmetric about the imaginary
axis, as will be done because XN is symmetric about this axis, the contribution from
the one stationary point equals the complex conjugate of the contribution from the
other so that both together give a real field with an amplitude that equals twice the
real part of Eq. (3.5.1). This completes the discussion about how to obtain the domi-
nant contributions to the transmitted field from a temporal sequence of graphs of the
transmittance and the method of steepest descent. In the next section, the electric field
contributions from the dominant stationary points will be numerically calculated.
48 The Brillouin precursor in photonic crystals
3.6 Results
Because the main interest is in the effect of the inhomogeneities of the multilayer on
the Brillouin precursor, only two simple input pulses are considered, namely a delta-
peak and a step-function modulated sinusoidal oscillation. For the delta-pulse, the
applied field is given by EL (t) = εδ(t) where ε is the strength of the pulse. In natural
time units, this applied field reads as
EL (θ) = εNδ(θ) , (3.6.1)
where εN = (Nl)−1cε is a strength with the dimension of an electric field amplitude.
For the input pulse of Eq. (3.6.1), Eq. (3.5.1) gives that the contribution to the trans-
mitted field from a stationary point at ω = φs (θ) equals
E(s)R (θ) =
NlεN
2πcΠ(θ;φs) . (3.6.2)
The field of Eq. (3.6.2), taken together with the contribution from the corresponding
stationary point at the opposite side of the imaginary axis, has been plotted in Fig. 3.5
for the three dominant stationary points. Shown are the individual contributions from
the stationary points and their sum, which approximately makes up for the Brillouin
precursor.
In order to be able to compare the amplitude of the transmitted Brillouin precursor
to that of an applied pulse, which is difficult to do for the delta-peak input pulse, we
also consider the Heaviside step-function modulated input signal
EL (t) = εθ(t)sinωct, (3.6.3)
where ε is the amplitude and ωc the carrier frequency and θ(t) is the unit step func-
tion. For this input field, Eq. (3.5.1) gives the contribution from a stationary point at
ω = φs to the transmitted field as
E(s)R (θ) =
ε
2π
ωc
ω2c −φ2
s
Π(θ;φs) . (3.6.4)
This field has been plotted in Fig. 3.6, with carrier frequency ωc = 4 · 1015s−1. The
amplitude is given in units ε and for the Brillouin precursor it is maximally about 0.6times that of the applied field. The amplitudes of applied and transmitted field are
still comparable because, for the choice of parameters of Table 3.4.1, the propagation
distance in the medium is 50nm and over such a short distance the absorption is very
small. Below, in the discussion section, we will give our reasons for taking such a
small propagation distance.
3.7 Discussion 49
ERHΘNL
s=I
s=II
s=III
1.4 1.6 1.8 2.0 2.2 2.4
0.0
0.5
1.0
1.5
2.0
2.5
ΘN
Ele
ctri
cfi
eld
inu
nit
sΕ N
Figure 3.5: Electric field amplitude (solid line) as a function of time, resulting from an
applied delta-peak input pulse that has been transmitted through the photonic crystal
with N = 1. The dashed lines give twice the real part of the individual contributions
from the three dominant stationary points.
3.7 Discussion
From Eq. (3.5.2), it follows that the instantaneous frequency of the contribution from
a stationary point in the complex frequency plane is approximately equal to the value
of the horizontal coordinate ξ of this point. Hence, at late times θ & 2, the transmit-
tance spectrum peaks at the resonance frequencies, whereas the Bragg-scattering fre-
50 The Brillouin precursor in photonic crystals
ERHΘNL
s=I
s=II
s=III
1.4 1.6 1.8 2.0 2.2 2.4
-0.4
-0.2
0.0
0.2
0.4
0.6
ΘN
Ele
ctri
cfi
eld
inu
nit
sΕ
Figure 3.6: Electric field amplitude (solid line) as a function of time, resulting from
an applied step-function modulated input pulse that has been transmitted through the
photonic crystal with N = 1. The dashed lines give twice the real part of the individual
contributions from the three dominant stationary points.
quency components that lie in between the resonance poles, are slightly suppressed.
In Fig. 3.5, it is visible that the contribution from the stationary point at ω = φI to
the transmitted field is exponentially decaying and non-oscillating, whereas the con-
tributions of the other stationary points are not only decaying, but also oscillating.
Our observation from Fig. 3.2 is that the number of stationary points of the phase
function and their locations in the complex plane are strongly tied up with the number
of singularities of this function and the locations of these singularities, a rigorous
proof is however lacking. As can be seen in Fig. 3.2, the locations of the singularities
of ΦN in the complex plane are independent of time. The singularities originate from
two possible causes. The slab permittivities εA and εB each have two poles in the
complex plane. From Eq. (3.2.2a), their locations are given by
ω = ±√
ω2m − γ2
m + iγm, m = A,B. (3.7.1)
The poles of the permittivities appear as branch-points of XN since this function
3.7 Discussion 51
depends on the square roots of the permittivities, see Eqs. (3.3.15). The poles of
Eq. (3.7.1) for m = A are located outside of the domain of the plots of Fig. 3.2 whereas
for m = B one is situated at ω ≃ (1−0.06i)ωB.
The other singularities of XN are the scattering resonance frequencies which
emerge in the landscape of XN as isolated poles. These poles are the roots of T22
(the 2,2-element of the transfer matrix, see Eq. (3.3.21)) and they appear in Fig. 3.2
as white dots. The figure shows that the scattering resonance poles cluster at the slab
permittivity pole. This is because of the following. The wavelength of the field in
slab m is given by λm = 2π/Rekm. For frequencies close to the pole of the permittivity
of this slab, the wavelength becomes very small and, as a consequence, many wave-
lengths fit in the slab giving many resonance poles, thus explaining the clustering.
The density of the scattering resonance poles increases with N, lA and lB since more
wavelengths fit in the system when the dimensions of the medium are increased. The
values of these geometry parameters were chosen small in order to keep the num-
ber of stationary points that give significant contributions to the Brillouin precursor
small: we only had to take into account three stationary point contributions.
There is also a serious drawback of taking the dimensions of the medium small.
As mentioned earlier, for our choice of parameters the medium width is only 50nm.
The phase function, and therewith XN , scales with these parameters. In the case of
propagation in a homogeneous medium, the propagation distance is a linear overall
scaling parameter of the exponent of the phase function. For an inhomogeneous
medium, the number of layers and the slab widths are not simple linear overall scaling
parameters, though the variation of XN does become small when the dimensions of
the medium are taken to be small. In the extreme case of having lA = 0 and lB = 0,
there is no variation at all since then XN ≡ 0 everywhere. The small variation of XN
in the complex plane is visible in Fig. 3.2 as the small difference of the values of XN
on neighboring contour lines, which is about 0.05. For larger medium dimensions,
the variation in XN becomes larger and the stationary point contributions are more
pronounced. This latter situation reflects the mature regime [123] of the dispersion,
in which the shape of the field has fully developed to a steady pattern. So, for the
parameters listed in Table 3.4.1, the Brillouin precursor has not yet fully reached the
steady state.
Fig. 3.7 is a contourplot of XN for N = 5 and θ = 1.3, again for the parameters of
Table 3.4.1. This figure shows three consequences of increasing the number of layers.
At first, the density of scattering resonance poles increases, which has been explained
above. Secondly, the resonance poles have shifted towards the real axis. The vertical
52 The Brillouin precursor in photonic crystals
XN for N= 5 at Θ=1.3
-1-0.5
polesband gap
ΦI
ΦII ΦIII
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
ΞΩB
ΗΩ
B
Figure 3.7: Contourplot of XN for N = 5 for the parameters listed in Table 3.4.1. The
main differences with the previous plots for N = 1 are explained in the text.
position of a pole in the complex plane is related to the contrast of the multilayer
at the frequency of the pole. This can easily be shown for a single slab of width lBwith constant, real index of refraction nB, surrounded by a homogeneous medium
with constant, real index of refraction nA. The Fresnel coefficients for reflection
inside the slab against the interfaces and for transmission through the interfaces to
the outside of the slab are respectively rBA = (nB −nA)/(nB +nA) and tBA = 1+ rBA.
The transmission coefficient of the slab equals
tslab = tABtBA exp(ilBnBω/c)/(1− r2
BA exp(2ilBnBω/c)). (3.7.2)
The scattering resonance is given by the zero of the denominator of tslab so that the
vertical coordinate of the pole follows as η = c(lBnB)−1log |rBA|. Since 0 < |rBA|< 1,
the pole lies below the real axis and approaches this axis if the contrast |rBA| ap-
proaches its maximum value of one. When the number of layers increases, the over-
all medium contrast increases as well and therefore the poles merge to the real axis.
3.8 Conclusions 53
The third consequence of increasing the number of layers is that the transmittance
minimum at the band gap becomes visible. It is located between the N-th and the
N +1-th resonance poles with nonzero frequency, at ω ≃ (0.57−0.05i)ωB.
At last, the Brillouin precursor that arises in the multilayer that consists of two
alternating homogeneous slabs with each one electron resonance is compared with
the Brillouin precursor that arises in a homogeneous medium with two electron res-
onances [97]. For comparison, the two electron resonances of the homogeneous
medium are taken to be equal to the two slab electron resonances of the inhomo-
geneous medium. In the case of the multilayer, we have found that the Brillouin
precursor is generated from the frequency components below the lowest electron res-
onance frequency of the two slabs and the forerunner is slightly distorted such that
the scattering resonance frequency components are enhanced and the Bragg-scattered
frequency components are suppressed, as has been concluded from the transmittance
landscape. In the case of the double-resonance homogeneous medium, the Brillouin
precursor is generated from the same low-frequency components, but the distortion
is absent [97].
3.8 Conclusions
We have investigated the electromagnetic Brillouin precursor that has been transmit-
ted through a one-dimensional photonic crystal, modeled by the stratified multilayer.
This precursor is formed by those components of the applied pulse that have frequen-
cies smaller than the lowest electron resonance frequency of the medium. From an
investigation of the transmittance of the multilayer and from applying the method
of steepest descent, the following observations have been made. The effect of the
slab contrast of the multilayer on the frequency spectrum of the transmitted Brillouin
precursor is that, after a certain rise time, the components with frequencies equal
to those of the scattering resonances of the multilayer are more pronounced and the
Bragg-scattering frequency components that lie in between the poles, are suppressed,
as compared to the frequency spectrum of the Brillouin precursor that has been trans-
mitted through a homogeneous medium with two electron resonances.
Chapter 4
Multilayer transmission coefficient
from a sum of light-rays
In this chapter1, the transmission coefficient of a multilayer for electromagnetic plane
waves is obtained by summing the transmission coefficients of the individual light-
rays in the multilayer. As compared to the transfer-matrix method, this derivation
results in a more intelligible expression. It turns out that the sum of all light-rays
through the multilayer forms a geometric series, exactly as in the case of a mono-
layer. The multilayer does not have to be periodic; the layers have arbitrary physical
lengths and arbitrary permittivities.
4.1 Introduction
The propagation of electromagnetic fields in layered, or piecewise homogeneous me-
dia, is intensively studied in the field of optics [1, 7, 14, 106]. The relatively recent
interest in photonic crystals [1] has renewed and enhanced the research of this sub-
ject.
The quantitative description of the transmission and reflection of electromagnetic
waves through/against multilayers is given by the transmission and reflection coef-
ficients of these media. These coefficients give the ratio of the electric field am-
plitudes of the transmitted or reflected wave to the electric field amplitude of the
incident wave. Usually, these coefficients are calculated with the transfer-matrix
1This chapter is based on R. Uitham and B. J. Hoenders, JEOS RP 3, 08013 (2008)
56 Multilayer transmission coefficient from a sum of light-rays
method [124], yielding rather complicated expressions, especially when the num-
ber of layers exceeds one. As will be shown in this chapter, a calculation of the
transmission coefficient of the multilayer as the sum of the transmission coefficients
of all individual light-rays in the multilayer results in a much simpler expression. It
turns out that, for a multilayer, the structure of the transmission coefficient remains
the same as it is for a monolayer. Key point in the calculation is the introduction of an
efficient basis for the possible paths along which the transmitted light-rays propagate.
For the transmitted field, this decomposition is rather straightforward, because the in-
dividual transmitted light-rays all have one single path in common. For the reflected
field, the situation is slightly more complicated. There is not one single common
path for the light-rays, since a light-ray can penetrate into the multilayer up to any
of the layers. Next to this, the light does not propagate along these common paths
in only one, but in both directions, since the light is eventually reflected. Because of
these slight complications (and because a lack of time), we have not yet derived the
reflection coefficient of the multilayer from summing the reflection coefficients of the
individual light-rays. It is however to be expected that the expression for the reflec-
tion coefficient will appear similarly simple as that for the transmission coefficient,
when it is calculated via the sum-over-all-light-rays.
This chapter has been organized as follows. First, the medium is modeled in
Sect. 4.2. Then, in Sect. 4.3, the amplitude coefficients that belong to the basic physi-
cal processes of the individual light-rays within the medium are given. In Sect. 4.4, a
basis is formed for the paths along which the transmitted light-rays propagate within
the medium. Sect. 4.5 gives the possible sequences in which these various basic path
elements can be taken by the light-rays. With the use of this basis and combinatorics,
the sum of all transmitted light-rays, and therewith the transmission coefficient, is
obtained in Sect. 4.6. A brief conclusion is given in Sect. 4.7.
4.2 Model for the medium
Our model for the multilayer has been depicted in Fig. 4.1. The response of the
system to an electromagnetic field varies stepwise along the x-axis. Including the
two homogeneous subspaces that bound the multilayer from the left and from the
right, there are in total N + 2 homogeneous subspaces. As has been indicated in
Fig. 4.1, these subspaces have been labeled as q = L,1, . . . ,N,R from left to right.
The positions of the interfaces i = 1, . . . ,N +1 that bound the subspaces are at x = xi
and the interfaces 1 and N +1 are respectively called the entrance and exit interfaces.
4.3 Electromagnetic field in the medium 57
L 1 2 · · · N R
x
y
⊙z
•
kL
θL
l1 l2 lN
εL ε1 ε2 · · · εN εR
µL µ1 µ2 · · · µN µR
x1 x2 x3 · · · xN xN+1
Figure 4.1: The multilayer. Layer q has physical width lq, and εq and µq are respec-
tively the permittivities and permeabilities of homogeneous subspace q. Also shown
is the wave-vector kL of the plane-wave incident field.
The subspaces 1 to N are the actual layers of the multilayer and the physical length
of layer q is equal to
lq = xq+1 − xq, (4.2.1)
where xq+1 denotes the coordinate of the interface immediately to the right of inter-
face q. The response of the multilayer and the two surrounding media to the elec-
tromagnetic field is taken to be causal, linear and isotropic and in each subspace it
is homogeneous. The analysis allows for dispersion and absorption, so the absolute
permittivity εq and absolute permeability µq in subspace q can be complex functions
of frequency. Now that the medium has been modeled, the effect of this medium on
an electromagnetic field will be analyzed in the following section.
4.3 Electromagnetic field in the medium
In this section, it will be shown how the amplitude of the electric field of a light-ray
is affected by the elementary physical processes that the light-ray can perform within
the multilayer system. These processes are propagation in the homogeneous layers,
and transmission through and reflection against the interfaces.
58 Multilayer transmission coefficient from a sum of light-rays
Electromagnetic fields are governed by Maxwell’s equations. In a medium that
does not contain free electric charges and currents, these equations read as
∇×E+ B = 0, (4.3.1a)
∇×H− D = 0, (4.3.1b)
∇ ·D = 0, (4.3.1c)
∇ ·B = 0, (4.3.1d)
where E is the electric field, B the magnetic induction and where the macroscopic
field quantities are the electric displacement D = D [E,B] and the magnetic field H =H [E,B]. In Fourier representation, with F ∈ E,H,D,B,
F(t,r) =∫
dωF(ω;r)exp(−iωt) , (4.3.2)
where
F(ω;r) =1
2π
∫dt F(t,r)exp(iωt) . (4.3.3)
For linear and isotropic media,
D = εE, (4.3.4a)
H = µ−1B, (4.3.4b)
where ε and µ are respectively the absolute permittivity and absolute permeability,
which can both be complex functions of ω. With Eqs. (4.3.4), Eqs. (4.3.1) lead to
(∇2 +k2
)E+(∇ lnµ)×∇× E+∇
(E ·∇ lnε
)= 0, (4.3.5a)
(∇2 +k2
)H+(∇ lnε)×∇× H+∇
(H ·∇ lnµ
)= 0, (4.3.5b)
where
k2 = ω2εµ. (4.3.6)
In the homogeneous subspaces q of Sect. 4.2, Eqs. (4.3.5) reduce to the Helmholtz
equation,
(∇2 +k2
q
)
Eq
Hq
= 0, (4.3.7)
where Eq and Hq denote the Fourier transformed fields in subspace q and k2q =
ω2εqµq. The plane of incidence is taken as the xy-plane and the applied field is a
4.3 Electromagnetic field in the medium 59
plane wave, incident from the left on the medium propagating in the rightwards di-
rection under an angle θ = θL with the x-axis, see Fig. 4.1. Snell’s law [14] implies
that in subspace q the wave-vector of the rightwards propagating field that results
from the applied field is given by
kq = xω
√εqµq − εLµL sin2 θL + yω
√εLµL sinθL, (4.3.8)
where, for q = L, this gives the wave-vector of the applied field. The square roots in
Eq. (4.3.8) are understood to have a positive real part. The TE-polarized, left- and
rightwards propagating electric field solutions of Eq. (4.3.7) are respectively given
by
E(l)q (ω;x,y) = zE
(l)q (ω;x,y) , (4.3.9a)
E(r)q (ω;x,y) = zE
(r)q (ω;x,y) , (4.3.9b)
with
E(l)q (ω;x,y) = A
(l)q (ω)exp(−ikq,x (x− xq)+ ikq,yy) , (4.3.10a)
E(r)q (ω;x,y) = A
(r)q (ω)exp(ikq,x (x− xq)+ ikq,yy) , (4.3.10b)
where kq,x and kq,y denote respectively the x- and y-components of kq. According to
Eqs. (4.3.1a) and (4.3.4b), the components of the magnetic fields that belong to the
TE-polarized solutions of Eqs. (4.3.9) are given by
H(l)q (ω;x,y) = x
kq,y
µqωE
(l)q (ω;x,y)+ y
kq,x
µqωE
(l)q (ω;x,y) , (4.3.11a)
H(r)q (ω;x,y) = x
kq,y
µqωE
(r)q (ω;x,y)+ y
−kq,x
µqωE
(r)q (ω;x,y) . (4.3.11b)
The coefficients A(l)q and A
(r)q in Eqs. (4.3.10) and (4.3.11) are determined by the
Fourier transforms of the tangential electric and magnetic fields at the interfaces as
A(l)q (ω) =
1
2
(Eq (ω;xq,0)+(µqω/kq,x) Hq,y (ω;xq,0)
), (4.3.12a)
A(r)q (ω) =
1
2
(Eq (ω;xq,0)− (µqω/kq,x) Hq,y (ω;xq,0)
), (4.3.12b)
60 Multilayer transmission coefficient from a sum of light-rays
where Hq,y is the y-component of the total magnetic field. From Eqs. (4.3.9) and (4.3.11),
it follows that continuity of the tangential electric and magnetic fields at interface q
gives respectively
πq−1A(l)q−1 +π−1
q−1A(r)q−1 = A
(l)q + A
(r)q , (4.3.13a)
kq−1,x
µq−1ω
(πq−1A
(l)q−1 −π−1
q−1A(r)q−1
)=
kq,x
µqω
(A
(l)q − A
(r)q
), (4.3.13b)
where we have introduced
πq ≡ exp(ikq,xlq) . (4.3.14)
The complex Fourier coefficient of the electric field in subspace q is given by
Eq =
√Eq · Eq. (4.3.15)
The Fresnel coefficients that belong to interface q are defined as
rq−1,q ≡(
E(l)q−1/E
(r)q−1
)∣∣∣E
(l)q =0
, (4.3.16a)
rq,q−1 ≡(
E(r)q /E
(l)q
)∣∣∣E
(r)q−1=0
, (4.3.16b)
tq−1,q ≡(
E(r)q /E
(r)q−1
)∣∣∣E
(l)q =0
, (4.3.16c)
tq,q−1 ≡(
E(l)q−1/E
(l)q
)∣∣∣E
(r)q−1=0
, (4.3.16d)
where all the Fourier coefficients are evaluated at x = xq and where q± 1 refers to
the medium to the left (−) or right (+) of medium q. Hence, from Eqs. (4.3.13) and
Eqs. (4.3.16), it follows that for TE-polarization and for q′ = q±1,
rqq′ =µ−1
q
√εqµq − εLµL sin2 θL −µ−1
q′
√εq′µq′ − εLµL sin2 θL
µ−1q
√εqµq − εLµL sin2 θL +µ−1
q′
√εq′µq′ − εLµL sin2 θL
, (4.3.17a)
tqq′ =2µ−1
q
√εqµq − εLµL sin2 θL
µ−1q
√εqµq − εLµL sin2 θL +µ−1
q′
√εq′µq′ − εLµL sin2 θL
, (4.3.17b)
4.3 Electromagnetic field in the medium 61
The admitted TM-or p-polarized magnetic field solutions of Eq. (4.3.7) that represent
respectively left- and rightwards propagating fields are given by
H(l)q (ω;x,y) = z B
(l)q (ω)exp(−ikq,x (x− xq)+ ikq,yy) , (4.3.18a)
H(r)q (ω;x,y) = z B
(r)q (ω)exp(ikq,x (x− xq)+ ikq,yy) , (4.3.18b)
From Eqs. (4.3.1b) and (4.3.4a), it follows that the components of the electric field
that belong to the TM-polarized solutions of Eqs. (4.3.18) are given by
E(l)q (ω;x,y) = x
−kq,y
εqωH
(l)q (ω;x,y)+ y
−kq,x
εqωH
(l)q (ω;x,y) , (4.3.19a)
E(r)q (ω;x,y) = x
−kq,y
εqωH
(r)q (ω;x,y)+ y
kq,x
εqωH
(r)q (ω;x,y) , (4.3.19b)
where H(l)q and H
(r)q are the Fourier coefficients of the left- and rightwards propa-
gating magnetic fields, these are defined similarly as those of the electric field in
Eq. (4.3.15). The coefficients B(l)q and B
(r)q in Eqs. (4.3.18) and (4.3.19) are deter-
mined by the Fourier transforms of the tangential electric and magnetic fields at the
interfaces as
B(l)q (ω) =
1
2
(Hq (ω;xq,0)− (εqω/kq,x) Eq,y (ω;xq,0)
), (4.3.20a)
B(r)q (ω) =
1
2
(Hq (ω;xq,0)+(εqω/kq,x) Eq,y (ω;xq,0)
). (4.3.20b)
From Eqs. (4.3.19) and (4.3.18), it follows that continuity of the tangential electric
and magnetic fields at interface q gives respectively
kq−1,x
εq−1ω
(πq−1B
(l)q−1 −π−1
q−1B(r)q−1
)=
kq,x
εqω
(B
(l)q − B
(r)q
), (4.3.21a)
πq−1B(l)q−1 +π−1
q−1B(r)q−1 = B
(l)q + B
(r)q , (4.3.21b)
and with Eqs. (4.3.16), it can be found that for TM-polarized fields, the Fresnel coef-
62 Multilayer transmission coefficient from a sum of light-rays
ficients take on the expressions
rqq′ =ε−1
q′
√εq′µq′ − εLµL sin2 θL − ε−1
q
√εqµq − εLµL sin2 θL
ε−1q
√εqµq − εLµL sin2 θL + ε−1
q′
√εq′µq′ − εLµL sin2 θL
, (4.3.22a)
tqq′ =
√εqµq′
εq′µq
2ε−1q
√εqµq − εLµL sin2 θL
ε−1q
√εqµq − εLµL sin2 θL + ε−1
q′
√εq′µq′ − εLµL sin2 θL
, (4.3.22b)
According to Eqs. (4.3.15), (4.3.9) and (4.3.19), the Fourier coefficients of the electric
field satisfy, both for TE- and for TM-polarization,
E(l)q (ω;xq,y) = πqE
(l)q (ω;xq+1,y) , (4.3.23a)
E(r)q (ω;xq+1,y) = πqE
(r)q (ω;xq,y) , (4.3.23b)
where πq has been defined in Eq. (4.3.14). Hence, propagation of a light-ray from
interface q+1 to interface q or vice versa gives, when the light-ray is evaluated at the
same y-positions at both interfaces, the electric field Fourier coefficient a factor πq.
Summarizing, the electromagnetic field is affected as follows by the multilayer.
Propagation within subspace q from interface q to interface q+1 or vice versa gives
the Fourier coefficient a factor πq of Eq. (4.3.14), which holds for both TE- and TM-
polarization. Reflection or transmission at interface q results in an factor given by
the Fresnel coefficients of Eqs. (4.3.17) for TE- and Eqs. (4.3.22) for TM-polarized
fields. The effects of the various elementary processes of propagation, reflection and
transmission on the Fourier coefficient of the electric field of a light-ray in interaction
with the multilayer have now been given, and in the following section the main part of
the work in this chapter begins. This is deriving an expression that gives all pathways
for the light-rays that contribute to the transmitted field.
4.4 Path decomposition
As a consequence of the reflections against interfaces there is an infinite number of
paths within the medium along which the applied field propagates from the entrance
to the exit interface. We have found a basis from which all these paths can be obtained
efficiently. The first step in the path decomposition is to observe that every continuous
path from the entrance to the exit interface of the system can be cut into two parts.
4.4 Path decomposition 63
One part of this path is the direct path, which is the continuous path straight from the
entrance to the exit interface of the medium. Along this path, there are no reflections
and the transmission coefficient for the light-ray that follows the direct path is equal
to
t(0)N = t01
N
∏q=1
tq,q+1πq, (4.4.1)
where tqq′ are the Fresnel transmission coefficients and πq the propagation factors
from the previous section. The other part in the decomposition of the path of a generic
transmitted light-ray consists of detour paths, these are the deviations from the direct
path. The transmission coefficient tN is the sum of the transmission coefficients of
the light-rays along all possible paths. Continuity of the paths implies that the path of
every possible transmitted light-ray always has, at least effectively, the direct path in
common. The transmission coefficient for the light-ray that follows the direct path,
Eq. (4.4.1), must therefore appear in the transmission coefficient of the multilayer as
a common factor,
tN = t(0)N δN , (4.4.2)
where δN denotes the factor that is equal to the sum of the amplitude coefficients of
the light-rays along all possible detour paths, the detour coefficient. The decompo-
sition into direct and detour paths has been illustrated in Fig. 4.2(a) for an arbitrary
transmitted light-ray through a multilayer medium with N = 4.
Now, a basis will be formed for the detour paths. Along each detour path, the
light-ray performs a sequence of translations between two interfaces. Each of these
translations is initiated by a reflection against an interface and the end of a translation
is either just before the following reflection or at the point where the light-ray contin-
ues its propagation along the direct path. The translation also includes transmission
through intermediate interfaces if the leftmost and the rightmost interfaces along the
translation are not neighboring ones. For the translation between a given pair of two
different interfaces there are two possibilities. Either it starts with a reflection against
the leftmost interface of the pair of interfaces, followed by a rightwards translation to
the rightmost interface of the pair, or it starts with a reflection against the rightmost
interface of the pair, followed by leftwards translation to the leftmost interface of
the pair. The associated amplitude coefficients of these reflection-induced right- and
64 Multilayer transmission coefficient from a sum of light-rays
∈ t4
=⊕
t(0)4
∈ δ4 ∈ δ4
(a)
=
λ13 λ45
ρ12 ρ45
λ12
ρ13
(b)
λ13 λ45
ρ12 ρ45
λ12
ρ13
=
l13
l12
l45
• ••
• ••
• •
(c)
=• ••
• ••
• •
(d)
Figure 4.2: Path decomposition of an arbitrary transmitted light-ray in a multilayer
with N = 4. Dots indicate the inclusion of Fresnel coefficients along the path. (a)
Decomposition into a direct path and detour paths. (b) Decomposition of detour
paths into reflection-induced translations. (c) Decomposition of oppositely directed
reflection-induced translations into loops. (d) Net result of the decomposition.
leftwards translations between the interfaces p and q > p, are given by respectively
ρpq = rp,p−1πp
n−1
∏s=p+1
ts−1,sπs, (4.4.3a)
λpq = rq−1,qπq−1
n−2
∏s=p
ts+1,sπs, (4.4.3b)
4.5 Path realizations for multiply-scattered, transmitted
light-rays 65
where rpp′ are the Fresnel reflection coefficients from the previous section. The de-
composition of the detour paths into reflection-induced left- and rightwards transla-
tions has been illustrated in Fig. 4.2(b), for the example path of Fig. 4.2(a). Since,
for a given transmitted light-ray, the propagation along the direct path accounts for
the translation through the medium from left to right, the net axial field translation
in the sum of all detour paths of this light-ray should be zero. In every detour
path, each reflection-induced translation in the leftwards direction between two in-
terfaces should therefore at some stage be, at least effectively, compensated with the
reflection-induced rightwards translation. The reflection-induced translations effec-
tively come in oppositely directed pairs, and a basis set for all detour paths can be
formed with these pairs. The pair of oppositely directed reflection-induced transla-
tions between interfaces p and q give the field an amplitude coefficient
lpq = ρpqλpq. (4.4.4)
Since these combined translations start and end on the same interface, their path
closes in the axial direction and we are actually considering a basis of loops. The set
of different loops in the multilayer is given by
lpq
N
=N+1
∑s=2
s−1
∑t=1
lts. (4.4.5)
This set contains N (N +1)/2 elements. The combination of the oppositely directed
pairs of reflection-induced translations into loops has been illustrated in Fig. 4.2(c),
for the example detour paths of Fig. 4.2(b). The net result of the steps that have
been illustrated in Figs. 4.2(a) to 4.2(c) is given in Fig. 4.2(d). With Eq. (4.4.5), the
basis elements of the detour paths have been identified as loops, or back-and-forth
scattering events between interfaces. In the following section, the number of possible
realizations of a light-path with different types of loops along it is obtained.
4.5 Path realizations for multiply-scattered, transmitted
light-rays
Since the various loops along a light-path can generally be performed in more than
one sequence, the next task is to find the number of possible realizations of a light-
path along which the various types of loops take place a given number of times.
66 Multilayer transmission coefficient from a sum of light-rays
Starting point is to observe that, if there would exist only one type of loop lpq in the
multilayer, all possible detour paths in the multilayer are generated by the expression
δN (lpq) = (1− lpq)−1 , (4.5.1)
which gives the well-known geometric series. In this case, when only one type of
loop is included, there is only one realization for the path of a transmitted light-ray
with a given number of loops along it. When more than one type of loop is included,
there can be more than one realization for the path of the transmitted light-ray with a
given number of loops along it, because permutations of different types of loops can
result in new paths.
Although the loops are the basis path elements, along the actual path of the light,
these loops are not necessarily fully completed one after another, see for instance
how the loops l13 and l12 are performed in the original path in Fig. 4.2, where l12 has
already started before l13 has been completed. The latter is finished only after the full
performance of l12, so it is as if l12 takes place ’within’ l13, the loops are nested and
in order to be able to speak about a sequence of loops, one has to be more specific.
Every loop lpq starts with a reflection at interface q and we say that it is performed,
though it is not yet completed, at the moment that the opposite reflection at interface
p has occurred. Within this convention, l13 takes place before l12 along the example
path of Fig. 4.2.
Consider a continuous path of a transmitted light-ray with two different types of
loops on it, type lpq and type lp′q′ , with (p,q) 6= (p′,q′). Without loss of generality,
we put p ≤ p′. Since every back-forth reflection must be initiated by a rightwards
propagating light-ray that impinges upon the rightmost interface of the loop, lpq can
be followed by lp′q′ only if p < q′. This requirement is always fulfilled because
p ≤ p′ < q′. The reverse order, lp′q′ followed by lpq, can only take place if p′ < q
which means that the loops should be located in spatially partly overlapping layers.
So if p′ < q, there are(
nm
)realizations for the path of a light-ray which performs m
loops of the one type and n−m of the other whereas there is only one realization if
p′ ≥ q. Therefore, the detour coefficient that results from detours composed solely of
these two loop types is given by
δN
(lpq, lp′q′
)=[1−(lpq + lp′q′
)]−1if p′ < q, (4.5.2a)
δN
(lpq, lp′q′
)= (1− lpq)
−1(1− lp′q′
)−1if p′ ≥ q, (4.5.2b)
as can be immediately verified by working out the terms in the generated series. This
completes the set of rules for combining different types of loops such that the correct
4.6 Transmission coefficient via sum of all possible paths 67
number of realizations for the light-paths follows. In the next section, these rules will
be applied to all loops present in the multilayer, resulting in the expression for the
transmission coefficient.
4.6 Transmission coefficient via sum of all possible paths
In this section, a recurrent relation for the detour coefficient δN that occurs in the
transmission coefficient of Eq. (4.4.2) will be obtained. In the trivial case of having
zero layers, there is only one interface between subspace L and R. Eq. (4.4.5) gives
that, in this system, the light-path cannot perform any loops. Therefore,
δ0 = 1. (4.6.1)
In the case of a single layer, Eq. (4.4.5) gives the single loop between the entrance
and exit interface, l12. The sum over all loops l12 is obtained from Eq. (4.5.1) as
δ1 = (1− l12)−1 . (4.6.2)
For the double-layer system, Eq. (4.4.5) gives the three loops l12, l13 and l23. The
sum of all possible allowed combinations of these three loops can be obtained in
two ways. The first way is to start with l12 and l23 and leave out l13. According to
Eq. (4.5.2), δ2 (l12, l23) = (1− (l12 + l23 (1− l12)))−1
. Now l13 should be added as a
loop that is located in spatially partly overlapping layers with both l12 and l23. This
gives, with Eq. (4.5.2), that the expression for the two-layer system with all loops
present is equal to
δ2 = (1− (l12 + l23 (1− l12)+ l13))−1 . (4.6.3)
The other way is to start with the spatially partly overlapping loops l12 and l13 in the
absence of l23. According to Eq. (4.5.2), δ2(l12, l13) = (1− (l12 + l13))−1
. Now l23
should be included as a loop that has no spatial overlap with l12 and as a loop that
does have spatial overlap with l13. This also results in Eq. (4.6.3). Similarly as in
the case of a one-layer medium, where one type of loop is generated to all orders
by Eq. (4.6.2), the expression that generates the three types of loops to all orders in
a two-layer medium, Eq. (4.6.3), generates a geometric series as well, but now this
series has the argument
L2 = l12 +(1− l12) l23 + l13. (4.6.4)
68 Multilayer transmission coefficient from a sum of light-rays
This means that, regarding its transmission, the two-layer medium with all different
types of loops can be described as a single-layer medium with one single type of loop
with corresponding amplitude coefficient given by Eq. (4.6.4). From layer-by-layer
addition with repeated application of the result that every two-layer system can be
represented by an equivalent one-layer system with only one type of loop present in
it, it follows that the detour path-factor of the N-layer system has the form
δN = (1−LN)−1 , (4.6.5)
where LN is the amplitude coefficient of the single loop in the equivalent one-layer
system. We will derive a recurrent relation for LN for N = 1,2, . . .. From Eqs. (4.6.1)
and (4.6.5) it follows that
L0 = 0. (4.6.6)
Under the addition of an N-th layer to a system with N − 1 layers, the N new loops
l1,N+1 to lN,N+1 emerge. Of these, l1,N+1 has partial spatial overlap with all other
loops, hence they should be combined as in Eq. (4.5.2), giving LN (LN−1, l1,N+1) =LN−1 + l1,N+1. Loop l2,N+1 has partial spatial overlap with all loops except for those
in L1, therefore LN (LN−1, l1,N+1, l2,N+1) = LN−1 + l1,N+1 + l2,N+1 (1−L1). From ap-
plying this up to and including the last new loop lN,N+1, it follows that
LN = LN−1 +N
∑m=1
(1−Lm−1) lm,N+1. (4.6.7)
The transmission coefficient is therefore given by
tN = t(0)N (1−LN)−1 , (4.6.8)
where t(0)N is the transmission coefficient of the light-ray that follows the direct path
and is given by Eq. (4.4.1) and where LN is the coefficient of the equivalent single-
layer system loop, given by Eq. (4.6.7). Note that the effective loop LN is multi-linear
in all the elementary loops in
lpq
N
.
4.7 Conclusion
We have calculated the transmission coefficient of the multilayer by summing the
transmission coefficients of all individual light-rays. It has turned out that, just as in
the case of a monolayer, the sum of all transmitted light-rays in the multilayer can
be captured in a geometrical series. The basic elements in this series are back-forth
reflections, or loops, between the various pairs of interfaces of the medium.
Chapter 5
Scattering from systems that do
not display one-to-one coupling of
modes
In this chapter1, the theory for scattering of electromagnetic waves is developed for
scattering objects for which the natural modes of the field inside the object do not
couple one-to-one with those outside the scatterer. Key feature of the calculation
of the scattered fields is the introduction of a new set of modes. As an example,
we calculate the reflected and transmitted fields generated by an electromagnetic
plane wave that impinges upon a multilayer slab of which the layers are stacked
perpendicular to the boundary planes.
5.1 Introduction
The analysis of scattering- and boundary value problems is a very important branch of
physics and it has been extensively explored since the eighteenth century [125, 126].
One of the requisites for the possibility to obtain analytical solutions in closed form
for these problems is that the pertinent scalar- or vectorial wave equation admits a
potential or refractive index such that this equation separates. Another requirement,
essential for the analytical solution of scattering- and boundary value problems, is
1This chapter is based on B. J. Hoenders, M. Bertolotti and R. Uitham, J. Opt. Soc. Am. A 24, No. 4,
1189-1200 (2007)
70 Scattering from systems that do not display one-to-one coupling of modes
that the geometry of the scatterer fits with the geometry of the separable potential or
refractive index. In the case of scattering from, for instance, a sphere or cylinder, the
potential or index of refraction should have a spherical or cylindrical dependence, i.e.
the level surfaces of the potential or refractive index must coincide with the boundary
surfaces. Then, the field is calculated employing the technique of eigenfunction ex-
pansions which are generated by the set of ordinary differential equations that result
from the separation of the original partial differential equation, viz. the scalar- or
vectorial wave equation [125–127].
The exactly solvable boundary value problems in mathematical physics all share
one property, namely that the boundaries of the various geometries involved fully
coincide with the coordinate surfaces of the various separable coordinate systems for
the wave equation. This is the case for the scattering of waves from, for instance, a
half-plane, a complete sphere, an ellipsoid or a cylinder filled with a homogeneous
medium. The boundaries of all these objects fully coincide with a separable coordi-
nate system. However, no simple theory exists for the calculation of fields that have
been scattered from media with so-called non-fitting boundaries. For the scattering
from, for example, a wedge or a half-sphere, the theory becomes much more difficult
and no simple theory exists if these objects consist of materials with respectively a
layered structure or a radially dependent refractive index. No exact theory exists in
the sense given above for the scattering from, for instance, an insect eye or an array
of rectangular protrusions (telegrapher’s surface), see Fig. 5.1. The solution of such
problems is not known in terms of a series of eigenfunctions with known coefficients.
(a) (b)
Figure 5.1: Model examples of objects in which there is no one-to-one coupling of
the internal and external field modes: (a) the two-dimensional insect eye and (b) the
telegrapher surface. Different colors indicate regions in which the refractive index
can take on different values. The light is supposed to enter from above.
5.2 Hybrid mode expansions 71
The absence of a simple theory for scattering from objects with non-fitting bound-
aries stems from the following observation: one natural mode of the field outside the
scattering medium no longer couples to one natural mode inside the scatterer but to
an, in general, infinite number of modes. This observation is corroborated analyzing
the results of the Wiener-Hopf technique [7], which is, for instance, used for the de-
scription of scattering of waves by a half-cylinder and connects an infinity of modes
outside and inside the scatterer. As an example that supports this statement, one could
think of a slab that is made from layers with different constant indices of refraction
such that the layers are perpendicular to the planes of the slab. Then, a plane wave
mode outside the slab couples to an infinity of modes for the layered medium, un-
like for the case if the layers had been oriented parallel to the boundary planes of the
slab. In the latter case, one plane wave mode outside couples to one plane wave mode
inside the multilayer.
This chapter shows, however, that with the introduction of new, different, sets of
modes (one set for the inside- and one set for the outside of the scattering medium),
it is again as if one outside mode couples to one inside mode, which is one of the
essential requirements for exactly solvable scattering problems. This property of the
new set of modes thus enables one to solve the scattering problem the same way as
for the usual exactly solvable problems. In particular, we will consider the scattering
of an electromagnetic wave by a multilayer slab, such that the layers are oriented
perpendicular to the boundary planes of the slab, viz. the layers are rotated over 90
degrees with respect to the orientation considered in the ordinary theory of layered
media.
A survey of the background theory of the mathematical results that have been
used in this chapter is provided in App. C.
5.2 Hybrid mode expansions
In this section, the general theory for the scattering of TE- or TM-polarized elec-
tromagnetic waves from a slab of which the response to the field varies along the
boundary surface is treated. Fig. 5.2 illustrates the type of medium we have in mind.
In following sections, specific examples of such media will be treated. In the absence
72 Scattering from systems that do not display one-to-one coupling of modes
εL,µL εM (y) ,µM (y) εR,µR
x
y
x = xL x = xR
d
Figure 5.2: A slab of which the permittivity εM and permeability µM vary along the
boundary surface. The electromagnetic waves are supposed to scatter from the slab
side-edges at x = xL and x = xR. The width of the slab is d.
of free electric charges and currents, Maxwell’s equations read as
∇×E+ B = 0, (5.2.1a)
∇×H− D = 0, (5.2.1b)
∇ ·D = 0, (5.2.1c)
∇ ·B = 0, (5.2.1d)
where E is the electric field, B the magnetic induction, D the displacement field and
H the magnetic field. Let G denote one of these fields. In Fourier representation,
G(t,r) =∫
dωG(ω;r)exp(−iωt) , (5.2.2a)
where G(ω;r) =1
2π
∫dtG(t,r)exp(iωt) . (5.2.2b)
For linear and isotropic media,
D = εE, (5.2.3a)
H = µ−1B, (5.2.3b)
where ε and µ are respectively the absolute permittivity and permeability, Eqs. (5.2.1)
give
(∇2 +ω2εµ
)E+(∇ lnµ)×∇× E+∇
(E ·∇ lnε
)= 0, (5.2.4a)
(∇2 +ω2εµ
)H+(∇ lnε)×∇× H+∇
(H ·∇ lnµ
)= 0. (5.2.4b)
5.2 Hybrid mode expansions 73
Denote
ηE = ε, (5.2.5a)
ηH = µ. (5.2.5b)
The media to the left- and to the right of the slab are homogeneous and, inside the
slab, the response functions vary across the boundary planes of the slab, hence
ηF (x,y) =
ηFL , x < xL,
ηFM (y) , xL < x < xR,
ηFR , x > xR
F = E,H, (5.2.6)
see Fig. 5.2. There is no variation in the response functions in the z direction, which
is the direction ”outside the plane of the paper”. For TE-polarization, E = E z and the
magnetic field follows from Eq. (5.2.1a) as
H =i
ωµ
(−x∂yE + y∂xE
). (5.2.7)
For TM-polarization, H = H z and the electric field follows from Eq. (5.2.1b) as
E =i
ωε
(x∂yH − y∂xH
). (5.2.8)
We will only consider either TE- or TM-polarization. Let F denote the amplitude of
the z-component of the field, i.e. F = E in case of TE-polarization and F = H in case
of TM-polarization. In the various regions of space, the field is labeled in accordance
with the response functions of Eq. (5.2.6) as
F =
FL, x < xL,FM, xL < x < xR,FR, x > xR.
(5.2.9)
Eqs. (5.2.4) give that (m = L,R)
[∂2
x +∂2y + k2
m
]Fm =0, (5.2.10a)
[∂2
x +∂2y +ω2εMµM −
(∂y lnηF
M
)∂y
]FM = 0, (5.2.10b)
where
k2m = ω2εmµm, (5.2.11a)
74 Scattering from systems that do not display one-to-one coupling of modes
and where we introduced
ηE = µ, (5.2.12a)
ηH = ε. (5.2.12b)
For Eqs. (5.2.10), mode solutions of the following forms are tried:
Fm
(f 2;x,y
)= φm
(f 2;x
)ψm
(f 2;y
), (5.2.13a)
FM
(f 2;x,y
)= φM
(f 2;x
)ψM
(f 2;y
). (5.2.13b)
Of these, the trial modes outside the slab are indeed solutions if
(∂2
x + f 2m
)φm
(f 2;x
)= 0, (5.2.14a)
(∂2
y + f 2)
ψm
(f 2;y
)= 0, (5.2.14b)
where f 2 is the separation constant and where
f 2m = k2
m − f 2. (5.2.15)
The trial modes inside the slab are solutions if they satisfy
(∂2
x + f 2L
)φM
(f 2;x
)= 0, (5.2.16a)
(∂2
y + f 2)
ψM
(f 2;y
)= KF
M
(f 2;y
), (5.2.16b)
where
KFM =
(k2
L −ω2εMµM
)ψM +
(∂y lnηF
M
)∂yψM. (5.2.17)
Our specific choice for KFM implies that, when media L and M are chosen equal,
i.e. when there is no scattering at the interface at x = xL, this corresponds to having
KFM = 0.
The hybrid mode expansion requires that two points on both boundary surfaces
at x = xL and at x = xR are designated, at which boundary conditions can be imposed
on ψm and ψM. The unit of length in the vertical (y−)direction is chosen such that
these points are given by y = 0 and y = 1, the scattering is supposed to take place
inside the interval 0 ≤ y ≤ 1. Let
(∂2
y + f 2)
ψ( j)m
(f 2;y
)= 0, j = 1,2, (5.2.18)
5.2 Hybrid mode expansions 75
with
ψ(1)m
(f 2;1
)= 1, (5.2.19a)
(∂yψ
(1)m
(f 2;y
))∣∣∣y=1
= 0, (5.2.19b)
ψ(2)m
(f 2;1
)= 0, (5.2.19c)
(∂yψ
(2)m
(f 2;y
))∣∣∣y=1
= 1. (5.2.19d)
The functions ψ(1,2)m are linearly independent solutions since their Wronskian,
W(
ψ(1)m ,ψ
(2)m
)= det
(ψ
(1)m ψ
(2)L
∂yψ(1)m ∂yψ
(2)m
), (5.2.20)
is equal to one. Note that dW/dy = 0, because of Eq. (5.2.18). The general solution
to Eq. (5.2.14b) is given by
ψm
(f 2;y
)= h1ψ
(1)m
(f 2;y
)+h2ψ
(2)m
(f 2;y
), (5.2.21)
with h1 and h2 arbitrary constants. Note that these constants do not depend on m, so
that we have chosen ψL = ψR. With Eq. (5.2.21), the boundary conditions on ψm are
given by
ψm
(f 2;1
)= h1, (5.2.22a)
(∂yψm
(f 2;y
))∣∣y=1
= h2. (5.2.22b)
With the auxiliary function
ψ(4)m
(f 2;y,y′
)= ψ
(1)m
(f 2;y
)ψ
(2)m
(f 2;y′
)−ψ
(2)m
(f 2;y
)ψ
(1)m
(f 2;y′
), (5.2.23)
and with a particular solution to the homogeneous differential equation, Eq. (5.2.14b),
given by
ψm
(f 2;y
)= βψ
(4)m
(f 2;0,y
)−α
(∂y′ψ
(4)m
(f 2;y′,y
))∣∣∣y′=0
, (5.2.24)
the solution to Eq. (5.2.16b) can be constructed as
ψM
(f 2;y
)=∫ y
0dy′KF
M
(f 2;y′
)ψ
(4)m
(f 2;y′,y
)+ψm
(f 2;y
), (5.2.25)
76 Scattering from systems that do not display one-to-one coupling of modes
The arbitrary constants α and β in Eq. (C.1.7) determine the boundary conditions of
ψM as
ψM
(f 2;0
)= ψm
(f 2;0
)= α, (5.2.26a)
(∂yψM
(f 2;y
))∣∣y=0
=(∂yψm
(f 2;y
))∣∣y=0
= β. (5.2.26b)
It can be calculated that
h1
(∂yψM
(f 2;y
))∣∣y=1
−h2ψM
(f 2;1
)
=∫ 1
0dyKF
M
(f 2;y
)ψm
(f 2;y
)+βψm
(f 2;0
)−α
(∂yψm
(f 2;y
))∣∣y=0
. (5.2.27)
With the definition
N(
f 2)
=∫ 1
0dyKF
M
(f 2;y
)ψm
(f 2;y
)+βψm
(f 2;0
)−α
(∂yψm
(f 2;y
))∣∣y=0
− γ,
(5.2.28)
where γ is a constant, Eq. (5.2.27) reads as
h1
(∂yψM
(f 2;y
))∣∣y=1
−h2ψM
(f 2;1
)= γ+N
(f 2). (5.2.29)
For the hybrid mode expansion, the following discretisation condition for the separa-
tion constant is used,
N(
f 2)
= 0. (5.2.30)
In App. C, it is shown that the functions ψm and ψM satisfy the following complete-
ness relation
δ(y− y′
)= ∑
n
ψm
(f 2n ;y)
ψM
(f 2n ;y′
)
N′ ( f 2n )
, (5.2.31)
where the sum is over those separation constants f 2n that satisfy Eq. (5.2.30), and
where N′ ( f 2n
)=(dN/d f 2
)∣∣f 2= f 2
n. Eq. (5.2.31) gives an expansion for the delta-
distribution in terms of the free space modes and the medium modes, therefore we
call it the hybrid mode expansion.
In App. C, we derive an expansion for the delta-distribution similar to Eq. (5.2.31),
but with solely free space modes involved. This expansion reads as
δ(y− y′
)= ∑
n
ψm
(f 2n ;y)
ψm
(f 2n ;y′
)
N′ ( f 2n )
. (5.2.32)
5.2 Hybrid mode expansions 77
The rightwards (r)- and leftwards (l) propagating field solutions to Eqs. (5.2.14) are,
in the pertinent mode expansions, in the various parts of space, given by
F(r/l)
m (x,y) = ∑n
ρ(r/l)m
(f 2n
)exp(±i fn,m (x− xm))ψm
(f 2n ;y), (5.2.33a)
F(r/l)
M (x,y) = ∑n
ρ(r/l)M
(f 2n
)exp(±i fn,L (x− xL))ψM
(f 2n ;y), (5.2.33b)
where the plus-(minus-)signs belong to the rightwards (leftwards) propagating fields
and where
f 2n,m = k2
m − f 2n , . (5.2.34)
From Eqs. (5.2.31) and (5.2.32), it follows that the spectral densities in Eqs. (5.2.33)
are given by
ρ(r/l)m
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
m
(xm,y′
)ψm
(f 2n ;y′
), (5.2.35a)
ρ(r/l)M
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
M
(xL,y
′)ψm
(f 2n ;y′
). (5.2.35b)
The unknown spectral densities ρ(l)L , ρ
(r/l)M and ρ
(r)R of the four scattered fields are
determined from the conditions of continuity of the tangential components of the
electric and magnetic fields at the two boundaries,(
F(l)
L + F(r)L
)∣∣∣x=xL
=(
F(l)
M + F(r)
M
)∣∣∣x=xL
, (5.2.36a)
1
ηFL
(∂xF
(l)L +∂xF
(r)L
)∣∣∣x=xL
=1
ηFM
(∂xF
(l)M +∂xF
(r)M
)∣∣∣x=xL
, (5.2.36b)
(F
(l)M + F
(r)M
)∣∣∣x=xR
=(
F(l)
R + F(r)
R
)∣∣∣x=xR
, (5.2.36c)
1
ηFM
(∂xF
(l)M +∂xF
(r)M
)∣∣∣x=xR
=1
ηFR
(∂xF
(l)R +∂xF
(r)R
)∣∣∣x=xR
. (5.2.36d)
Eqs. (5.2.36b) and (5.2.36d) were obtained from Eqs. (5.2.7) and (5.2.8). The con-
ditions of Eq. (5.2.36) lead to a set of four coupled linear integral equations for the
unknown densities.
The conditions of Eq. (5.2.36) do not lead to a simple system of equations, be-
cause, as follows from inserting the fields of Eq. (5.2.33), both types of modes ψm
and ψM are involved. It is only in the case that these modes are identical2, that the
2Recall that this is the case for for instance the scattering from a homogeneous slab.
78 Scattering from systems that do not display one-to-one coupling of modes
spectral densities can be equated for each mode. But, this much desired property is
obtained from the hybrid mode expansion of Eq. (5.2.31), which allows for the fol-
lowing rewriting of the mode expansions for the fields inside the slab. Starting from
Eq. (5.2.33b), with Eq. (5.2.35b), we have that
F(r/l)
M (x,y) = ∑n
(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
M
(xL,y
′)ψm
(f 2n ;y′
)
× exp(±i fn,L (x− xL))ψM
(f 2n ;y). (5.2.37)
From Eq. (5.2.34), the differential equation for ψM (Eq. (5.2.16b)) and (5.2.17) it
follows that
fn,LψM
(f 2n ;y)
=√
∂2y + εM (y)µM (y)ω2 −
(∂y lnηF
M (y))
∂yψM
(f 2n ;y). (5.2.38)
Hence, Eq. (5.2.37) can be written as
F(r/l)
M (x,y) = exp
(±i(x− xL)
ö2
y + εM (y)µM (y)ω2 −(∂y lnηF
M (y))
∂y
)
×∫ 1
0dy′F(r/l)
M
(xL,y
′)∑n
(N′ ( f 2
n
))−1ψm
(f 2n ;y′
)ψM
(f 2n ;y). (5.2.39)
From the hybrid mode expansion of Eq. (5.2.31), which is symmetric in y and y′, it
follows that Eq. (5.2.39) must be equal to
F(r/l)
M (x,y) = exp
(±i(x− xL)
ö2
y + εM (y)µM (y)ω2 −(∂y lnηF
M (y))
∂y
)
×∫ 1
0dy′F(r/l)
M
(xL,y
′)∑n
(N′ ( f 2
n
))−1ψM
(f 2n ;y′
)ψm
(f 2n ;y). (5.2.40)
Now, the derivatives act on ψm and from using the pertinent differential equation,
Eq. (5.2.14b), it follows that Eq. (5.2.40) equals
F(r/l)
M (x,y) = ∑n
σ(r/l)M
(f 2n
)exp(±i fn,M (y)(x− xL))ψm
(f 2n ;y), (5.2.41)
where
fn,M (y) =√
εM (y)µM (y)ω2 − i fn
(∂y lnηF
M (y))− f 2
n (5.2.42)
5.2 Hybrid mode expansions 79
and with now
σ(r/l)M
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
M
(xL,y
′)ψM
(f 2n ;y′
). (5.2.43)
It can be verified that, in the representation of Eq. (5.2.41), F(r/l)
M satisfies Eq. (5.2.10b).
The conditions of Eq. (5.2.36) are then applied to the fields of Eqs. (5.2.33a) and (5.2.41).
Equating the coefficients of the modes ψm
(f 2n ;y) at both sides of the interface at
x = xL leads to3
∆L
(f 2n
)(
ρ(r)L
(f 2n
)
ρ(l)L
(f 2n
))
= ∆M
(f 2n ;y)(
σ(r)M
(f 2n
)
σ(l)M
(f 2n
))
(5.2.44)
where the dynamical matrices are given by
∆m
(f 2)
=
(1 1fm
ηFm
− fm
ηFm
), (5.2.45)
∆M
(f 2;y
)=
(1 1
fM(y)
ηFM(y)
− fM(y)
ηFM(y)
). (5.2.46)
Equating the coefficients of the modes ψm
(f 2n ;y′
) at both sides of the interface at
x = xR gives
∆M
(f 2n ;y′
)PM
(f 2n ;y′
)(
σ(r)M
(f 2n
)
σ(l)M
(f 2n
))
= ∆R
(f 2n
)(
ρ(r)R
(f 2n
)
ρ(l)R
(f 2n
))
, (5.2.47)
where d = xR − xL and where the propagation matrix is given by
PM
(f 2;y
)= diag(exp(i fM (y)d) ,exp(−i fM (y)d)) . (5.2.48)
Eqs. (5.2.44) and (5.2.47) can be solved for the unknown spectral densities of the
scattered fields using Cramer’s rule. Thus, one finds the inhomogeneous equivalents
3Note that the boundary conditions of Eqs. (5.2.44) and (5.2.47) reduce to the boundary conditions
for a homogeneous slab when εM and µM do not depend on y.
80 Scattering from systems that do not display one-to-one coupling of modes
of the Fresnel transmission and reflection coefficients for the interface between ho-
mogeneous medium L and inhomogeneous medium M respectively as
tLM
(f 2n ;y)≡(
σ(r)M
(f 2n
)
ρ(r)L ( f 2
n )
)∣∣∣∣∣σ
(l)M ( f 2
n )=0
=2(
fn,L/ηFL
)(
fn,L/ηFL
)+(
fn,M (y)/ηFM (y)
) , (5.2.49a)
rLM
(f 2n ;y)≡(
ρ(l)L
(f 2n
)
ρ(r)L ( f 2
n )
)∣∣∣∣∣σ
(l)M ( f 2
n )=0
=
(fn,L/ηF
L
)−(
fn,M (y)/ηFM (y)
)(
fn,L/ηFL
)+(
fn,M (y)/ηFM (y)
) , (5.2.49b)
Similar coefficients are obtained for the interface at x = xR. This gives the y-dependent
transmission and reflection coefficients of the complete slab for the rightwards prop-
agating waves of Eq. (5.2.33a) respectively as
tM(
f 2n ;y)≡(
ρ(r)R
(f 2n
)
ρ(r)L ( f 2
n )
)∣∣∣∣∣ρ
(l)R ( f 2
n )=0
=tLM
(f 2n ;y)
tMR
(f 2n ;y)
exp(i fn,M (y)d)
1− rML ( f 2n ;y)rMR ( f 2
n ;y)exp(2i fn,M (y)d), (5.2.50a)
rM
(f 2n ;y)≡(
ρ(l)L
(f 2n
)
ρ(r)L ( f 2
n )
)∣∣∣∣∣ρ
(l)R ( f 2
n )=0
= rLM
(f 2n ;y)+
tLM
(f 2n ;y)
tML
(f 2n ;y)
rMR
(f 2n ;y)
exp(2i fn,M (y)d)
1− rML ( f 2n ;y)rMR ( f 2
n ;y)exp(2i fn,M (y)d).
(5.2.50b)
The reflected and transmitted fields that result from a field F(r)
L that is applied from
the left to the slab follow respectively as
F(l)
L (x,y) = ∑n
rM
(f 2n ;y)
ρ(r)L
(f 2n
)exp(−i fn,L (x− xL))ψm
(f 2n ;y), (5.2.51a)
F(r)
R (x,y) = ∑n
tM(
f 2n ;y)
ρ(r)L
(f 2n
)exp(i fn,R (x− xR))ψm
(f 2n ;y). (5.2.51b)
5.2 Hybrid mode expansions 81
5.2.1 Modes in the rotated multilayer slab
As a specific example of the scattering geometry that has been depicted in Fig. 5.2, we
consider a multilayer which consists of N periods of alternating slabs σ = A,B with
thicknesses lσ, permittivities εσ and permeabilities µσ. The geometry is depicted
in Fig. 5.3. The electromagnetic field is supposed to enter the medium through its
λ = 0
λ = 1
...
...
...
λ = N
λ = N +1
εL,µL
εA,µA
εA,µA lA
εA,µA
εA,µA
εA,µA
εB,µB lB
εB,µB
εB,µB
εR,µR
x
y
x = xL x = xR
y = yA0 = 0
y = yA1
y = yB1
y = yAN
y = yBN
y = yAN+1
y = 1
Figure 5.3: The rotated multilayer. The width, permittivity and permeability of slab
σ = A,B are respectively lσ, εσ and µσ and the yσλ denote the coordinates of the
interfaces between the slabs.
boundaries at x = xL, or/and x = xR. Therefore, as compared to its conventional ori-
entation with respect to the applied field, the multilayer of Fig. 5.3 is rotated over 90
degrees in the xy-plane. The right- and leftwards propagating parts of the field inside
82 Scattering from systems that do not display one-to-one coupling of modes
the medium are given, in their expansions in the free space modes, by Eq. (5.2.41),
F(r/l)
M (x,y) = ∑n
σ(r/l)M
(f 2n
)exp(±i fn,M (y)(x− xL))ψm
(f 2n ;y), (5.2.52)
where the spectral densities are given by
σ(r/l)M
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
M
(xL,y
′)ψM
(f 2n ;y′
), (5.2.53)
and where
fn,M (y) = fn,σ if yσλ ≤ y < yσλ + lσλ, (5.2.54)
with yσλ the interface coordinates, see Fig. 5.3. We further defined lA0 = yA1, lAN+1 =1− yAN+1 and lσλ = lσ if λ = 1, . . . ,N. The modes inside the medium, for yσλ ≤ y ≤yσλ + lσλ, are given by
ψM
(f 2;y
)= χ
(u)σλ exp(i fσL (y− yσλ))+χ
(d)σλ exp(−i fσL (y− yσλ)) , (5.2.55)
with fσL =√
k2σ − f 2
L . The coefficients χ(u)σλ and χ
(d)σλ of respectively the up- and
downwards propagating parts of the modes in Eq. (5.2.55) satisfy [117]
(χ
(u)B1
χ(d)B1
)= ΘBAPA
(χ
(u)A1
χ(d)A1
)(5.2.56a)
(χ
(u)σλ
χ(d)σλ
)=
(ελ−1 − ε1−λ
ε− ε−1Tσ −
ελ−2 − ε2−λ
ε− ε−1
)(χ
(u)σ1
χ(d)σ1
), (5.2.56b)
where the transmission and propagation matrices are respectively given by
ΘBA =1
2
(1+ηAB 1−ηAB
1−ηAB 1+ηAB
), ηAB ≡ ηB
ηA
fAL
fBL
, (5.2.57)
Pσ = diag(exp(i fσLlσ) ,exp(−i fσLlσ)) , (5.2.58)
and where ε = 12trTσ +
√(12trTσ
)2 −1 and ε−1 are the eigenvalues of the single-layer
transfer matrix,
Tσ =
(Aσ Bσ
Cσ Dσ
), (5.2.59)
5.2 Hybrid mode expansions 83
for transfer between successive slabs σ. The entries of this matrix are given by [106,
116]
Aσ = exp(i fσLlσ)
[cos( fσLlσ)+
i
2
(ηF
σ
ηFσ
fσL
fσL
+ηF
σ
ηFσ
fσL
fσL
)sin( fσLlσ)
], (5.2.60a)
Bσ =i
2exp(−i fσLlσ)
(ηF
σ
ηFσ
fσL
fσL
− ηFσ
ηFσ
fσL
fσL
)sin( fσLlσ) , (5.2.60b)
Cσ =−i
2exp(i fσLlσ)
(ηF
σ
ηFσ
fσL
fσL
− ηFσ
ηFσ
fσL
fσL
)sin( fσLlσ) , (5.2.60c)
Dσ = exp(−i fσLlσ)
[cos( fσLlσ)− i
2
(ηF
σ
ηFσ
fσL
fσL
+ηF
σ
ηFσ
fσL
fσL
)sin( fσLlσ)
], (5.2.60d)
where σ = B if σ = A and σ = A if σ = B. The modes outside and inside the medium
fulfil Eq. (5.2.31) if they satisfy the boundary conditions
ψm
(f 2;1
)= h1, (5.2.61a)
(∂yψm
(f 2;y
))∣∣y=1
= h2, (5.2.61b)
and
ψM
(f 2;0
)= α, (5.2.62a)
(∂yψM
(f 2;y
))∣∣y=0
= β, (5.2.62b)
and if the separation constant f 2 satisfies the discretisation condition N(
f 2)
= 0,
with N from Eq. (C.1.10). This condition implies
h1
(∂yψM
(f 2n ;y))∣∣
y=1−h2ψM
(f 2n ;1)
= γ. (5.2.63)
Eq. (5.2.32) is obtained from the auxiliary set of free space modes ψm
(f 2n ;y), that
satisfy the ‘medium boundary conditions’,
ψm
(f 2;0
)= α, (5.2.64a)
(∂yψm
(f 2;y
))∣∣y=0
= β. (5.2.64b)
The appropriate set of modes ψM
(f 2n ;y) has now been defined for the rotated mul-
tilayer, so that the scattering theory developed above can be applied to this particular
system.
84 Scattering from systems that do not display one-to-one coupling of modes
5.2.2 Scattering from a semi-infinite line
The theory developed above will first be applied to the simple example of a medium
with only one boundary surface and only one layer. This medium has been depicted
in Fig. 5.4. The response functions are given by
ηF (x,y) =
ηF
0 , x ≤ 0,ηF
0
(1+ lδ(y− y′)χF
M
), x > 0,
(5.2.65a)
ηF (x,y) = ηF0 . (5.2.65b)
where ηE0 = ηH
0 = ε0, ηH0 = ηE
0 = µ0, l is a constant with the dimension of length and
χFM =
ηFM −ηF
0
ηF0
(5.2.66)
are the susceptibilities of the delta-peak. The boundary surface of the medium is at
x = 0, and the medium is infinitely extended in the direction of the positive x-axis. For
ε0,µ0
ε0,µ0
ε0,µ0
ηF0
(1+χF
Mlδ(y− y′)),ηF
0
x
y
x = 0 x = ∞
y = 0
y = y′
y = 1
Figure 5.4: The semi-infinite line.
the medium response of Eqs. (5.2.65), the driving force term of Eq. (5.2.17) becomes
KFM
(f 2;y
)= −k2
0χFMlδ
(y− y′
)ψM
(f 2;y
), (5.2.67)
where k20 = ε0µ0ω2. The solutions ψM to Eq. (5.2.16b) that satisfy the boundary
conditions of Eqs. (5.2.26) are given by Eq. (5.2.25). With Eq. (5.2.67), one finds
5.2 Hybrid mode expansions 85
that
ψM
(f 2;y
)=
ψL
(f 2;y
)if y < y′,
ψL
(f 2;y
)+ k2
0χFMlψM
(f 2;y′
)ψ
(4)L
(f 2;y,y′
)if y ≥ y′.
(5.2.68)
From putting y = y′ in the latter equation, it follows that ψM
(f 2;y′
)= ψL
(f 2;y′
),
hence
ψM
(f 2;y
)=
ψL
(f 2;y
)if y < y′,
ψL
(f 2;y
)+ k2
0χFMlψL
(f 2;y′
)ψ
(4)L
(f 2;y,y′
)if y ≥ y′.
(5.2.69)
We consider the scattering of a rightwards propagating electromagnetic field F(r)
L
at x < 0 that enters the medium of Fig. 5.4 from the left-hand-side. The structure
gives rise to a leftwards propagating reflected field F(l)
L at x < 0, and a rightwards
propagating ’transmitted’ field F(r)
M at x > 0. The expansions of the fields into the
pertinent modes are
F(r/l)
L (x,y) = ∑n
ρ(r/l)L
(f 2n
)exp(±i fn,Lx)ψL
(f 2n ;y), (5.2.70a)
F(r)
M (x,y) = ∑n
ρ(r)M
(f 2n
)exp(i fn,Lx)ψM
(f 2n ;y), (5.2.70b)
where the spectral densities are given by
ρ(r/l)L
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′′F(r/l)
L
(0,y′′
)ψL
(f 2n ;y′′
), (5.2.71a)
ρ(r)M
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′′F(r)
M
(0,y′′
)ψL
(f 2n ;y′′
). (5.2.71b)
The boundary conditions at x = 0 give that
∑n
(ρ
(l)L
(f 2n
)+ρ
(r)L
(f 2n
))ψL
(f 2n ;y)
= ∑n
ρ(r)M
(f 2n
)ψM
(f 2n ;y), (5.2.72a)
∑n
fn,L
(ρ
(r)L
(f 2n
)−ρ
(l)L
(f 2n
))ψL
(f 2n ;y)
= ∑n
fn,Lρ(r)M
(f 2n
)ψM
(f 2n ;y). (5.2.72b)
With ε ≡ lk20χF
M , the modes in the medium take on the form
ψM
(f 2n ;y)
= ψL
(f 2n ;y)+ εψ
(1)M
(f 2n ;y), (5.2.73)
86 Scattering from systems that do not display one-to-one coupling of modes
where the function
ψ(1)M
(f 2n ;y)
= ψL
(f 2n ;y′
)ψ
(4)L
(f 2n ;y,y′
)(5.2.74)
denotes the perturbed part of the modes at x > 0. Eqs. (5.2.72) will be solved in a
perturbative manner, considering ε to be such that the magnitude of the disturbance∣∣∣εψ(1)M
∣∣∣ is small as compared to |ψL|. According to perturbation theory, the scattered
fields are written, to first order in ε, as
ρ(l)L = ρ
(l,0)L + ερ
(l,1)L , (5.2.75a)
ρ(r)M = ρ
(r,0)M + ερ
(r,1)M , (5.2.75b)
Eqs. (5.2.72) give for the terms at zeroth order in ε that
∑n
(ρ
(l,0)L
(f 2n
)+ρ
(r)L
(f 2n
))ψL
(f 2n ;y)
= ∑n
ρ(r,0)M
(f 2n
)ψL
(f 2n ;y), (5.2.76a)
∑n
fn,L
(ρ
(r)L
(f 2n
)−ρ
(l,0)L
(f 2n
))ψL
(f 2n ;y)
= ∑n
fn,Lρ(r,0)M
(f 2n
)ψL
(f 2n ;y). (5.2.76b)
From equating the coefficients of the modes
ψL
(f 2n ;y)
, we obtain for the unper-
turbed spectral densities the following equations,
ρ(l,0)L
(f 2n
)+ρ
(r)L
(f 2n
)= ρ
(r,0)M
(f 2n
), (5.2.77a)
ρ(r)L
(f 2n
)−ρ
(l,0)L
(f 2n
)= ρ
(r,0)M
(f 2n
), (5.2.77b)
which has as obvious solution ρ(r,0)M
(f 2n
)= ρ
(r)L
(f 2n
)and ρ
(l,0)L
(f 2n
)= 0. To zeroth
order in ε, there is no scattering from the delta-peak. For the terms at first order in ε,
Eq. (5.2.72) gives that
∑n
ρ(l,1)L
(f 2n
)ψL
(f 2n ;y)
= ∑n
ρ(r,1)M
(f 2n
)ψL
(f 2n ;y)
+∑n
ρ(r,0)M
(f 2n
)ψ
(1)M
(f 2n ;y), (5.2.78a)
−∑n
fn,Lρ(l,1)L
(f 2n
)ψL
(f 2n ;y)
= ∑n
fn,Lρ(r,1)M
(f 2n
)ψL
(f 2n ;y)
+∑n
fn,Lρ(r,0)M
(f 2n
)ψ
(1)M
(f 2n ;y). (5.2.78b)
5.2 Hybrid mode expansions 87
The function ψ(1)M
(f 2n ;y)
reads, expanded in the medium modes, as
ψ(1)M
(f 2n ;y)
= ∑m
ρ(1)M
(f 2n , f 2
m
)ψM
(f 2m;y)
= ∑m
ρ(1)(0)M
(f 2n , f 2
m
)ψL
(f 2m;y)+O (ε) , (5.2.79)
with
ρ(1)M
(f 2n , f 2
m
)=(N′ ( f 2
m
))−1∫ 1
0dy′′ψ(1)
M
(f 2n ;y′′
)ψL
(f 2m;y′′
), (5.2.80a)
ρ(1)(0)M
(f 2n , f 2
m
)=(N′
0
(f 2m
))−1∫ 1
0dy′′ψ(1)
M
(f 2n ;y′′
)ψL
(f 2m;y′′
). (5.2.80b)
The last terms of Eqs. (5.2.78) are therefore, up to terms at higher order in ε, respec-
tively equal to
∑n
ρ(r,0)M
(f 2n
)ψ
(1)M
(f 2n ;y)
= ∑n
ρ(r,0)M
(f 2n
)∑m
ρ(1)(0)M
(f 2n , f 2
m
)ψL
(f 2m;y),
(5.2.81a)
∑n
fn,Lρ(r,0)M
(f 2n
)ψ
(1)M
(f 2n ;y)
= ∑n
fn,Lρ(r,0)M
(f 2n
)∑m
ρ(1)(0)M
(f 2n , f 2
m
)ψL
(f 2m;y).
(5.2.81b)
Hence, after interchanging the summation indices m and n in Eqs. (5.2.81) and with
ρ(r,0)M = ρ
(r)L from above, equating the coefficients of ψL
(f 2n ;y)
in the boundary con-
dition equations at first order in ε, results in
ρ(l,1)L
(f 2n
)= ρ
(r,1)M
(f 2n
)+∑
m
ρ(1)(0)M
(f 2m, f 2
n
)ρ
(r)L
(f 2m
), (5.2.82a)
− fn,Lρ(l,1)L
(f 2n
)= fn,Lρ
(r,1)M
(f 2n
)+∑
m
fm,Lρ(1)(0)M
(f 2m, f 2
n
)ρ
(r)L
(f 2m
). (5.2.82b)
Solving this for the unknown spectral densities, gives
ρ(l,1)L
(f 2n
)= ∑
m
fn,L − fm,L
2 fn,Lρ
(r)L
(f 2m
)ρ
(1)(0)M
(f 2m, f 2
n
), (5.2.83)
ρ(r,1)M
(f 2n
)= −∑
m
fn,L + fm,L
2 fn,Lρ
(r)L
(f 2m
)ρ
(1)(0)M
(f 2m, f 2
n
), (5.2.84)
88 Scattering from systems that do not display one-to-one coupling of modes
The overlap integral in ρ(1)(0)M
(f 2m, f 2
n
)can be readily evaluated, noting that
ψ(1)L
(f 2;y
)= cos( f (y−1)) , (5.2.85)
ψ(2)L
(f 2;y
)= f−1 sin( f (y−1)) , (5.2.86)
and hence the integrand contains trigonometric functions only:
ψL
(f 2n ;y′′
)= h1 cos
(fn
(y′′−1
))+h2 f−1
n sin(
fn
(y′′−1
)), (5.2.87)
ψ(1)M
(f 2m;y′′
)= f−1
m
(h1 cos
(fm
(y′−1
))+h2 f−1
m sin(
fm
(y′−1
)))sin(
fm
(y′− y′′
)).
(5.2.88)
5.2.3 Scattering from a layer with finite width
In this last section, we calculate the fields that are scattered from an interface between
a homogeneous medium and a medium with one layer with two slabs that have finite
widths. The geometry has been depicted in Fig. 5.5. As before, the incoming(
F(r)
L
),
εL,µL
ε1,µ1
ε2,µ2 l1
ε3,µ3
ε1,µ1
l2
x
y
x = xL x = ∞
y = y0 = 0
y = y1
y = y2
y = y3
y = y4 = 1
Figure 5.5: Geometry used for the numerical calculation.
reflected(
F(l)
L
)and transmitted
(F
(r)M
)fields are expressed in their mode expansions,
5.2 Hybrid mode expansions 89
F(r/l)
L (x,y) = ∑n
ρ(r/l)L
(f 2n
)exp(±i fn,Lx)ψL
(f 2n ;y), (5.2.89a)
F(r)
M (x,y) = ∑n
ρ(r)M
(f 2n
)exp(i fn,Lx)ψM
(f 2n ;y), (5.2.89b)
where
ρ(r/l)L
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r/l)
L
(0,y′)
ψL
(f 2n ;y′
), (5.2.90a)
ρ(r)M
(f 2n
)=(N′ ( f 2
n
))−1∫ 1
0dy′F(r)
M
(0,y′)
ψL
(f 2n ;y′
). (5.2.90b)
Continuity of the tangential components of the electric and magnetic fields at x = 0
gives for the various fields F that
(F
(r)L + F
(l)L
)∣∣∣x=0
= F(r)
M
∣∣∣x=0
, (5.2.91a)
1
ηFL
(∂x
(F
(r)L + F
(l)L
))∣∣∣x=0
=1
ηFR
(∂xF
(r)M
)∣∣∣x=0
. (5.2.91b)
Eq. (5.2.91b) implies, with Eqs. (5.2.89), that
(1/ηF
L
)∑n
fn,L
(ρ
(r)L
(f 2n
)−ρ
(l)L
(f 2n
))ψL
(f 2n ;y)=(1/ηF
M
)∑n
fn,Lρ(r)M
(f 2n
)ψM
(f 2n ;y).
(5.2.92)
Eq. (5.2.92) is expressed on the ψM-basis. Hereto, first expand the mode densities.
For the left-hand-side of Eq. (5.2.92), this gives
(1/ηF
L
)∑n
fn,L
(ρ
(r)L
(f 2n
)−ρ
(l)L
(f 2n
))ψL
(f 2n ;y)
=(1/ηF
L
)∫dy′(
F(r)
L
(0,y′)− F
(l)L
(0,y′))
∑n
ψL
(f 2n ;y)
fn,LψM
(f 2n ;y′
)
N′ ( f 2n )
, (5.2.93)
and for the right-hand-side of Eq. (5.2.92),
(1/ηF
M
)∑n
fn,Lσ(r)M
(f 2n
)ψM
(f 2n ;y)
=(1/ηF
M
)∫dy′F(r)
M
(0,y′)∑n
ψL
(f 2n ;y′
)fn,LψM
(f 2n ;y)
N′ ( f 2n )
. (5.2.94)
90 Scattering from systems that do not display one-to-one coupling of modes
Then, note that Eqs. (5.2.15), (5.2.16b) and (5.2.17) imply that
fn,LψM
(f 2n ;y′
)=√
∂2y′ +ω2εM (y′)µM (y′)−
(∂y′ lnηF
M (y′))
∂y′ψM
(f 2n ;y′
).
(5.2.95)
Eq. (5.2.31) implies
∑n
ψL
(f 2n ;y′
)ψM
(f 2n ;y)
N′ ( f 2n )
= ∑n
ψL
(f 2n ;y)
ψM
(f 2n ;y′
)
N′ ( f 2n )
. (5.2.96)
The square root operator in Eq. (5.2.95) can thus be switched to act on ψL instead of
ψM. With using Eq. (5.2.14b), it follows that
ö2
y′ +ω2εM (y′)µM (y′)−(∂y′ lnηF
M (y′))
∂y′ψL
(f 2n ;y′
)
=√
ω2εM (y′)µM (y′)− i fn
(∂y′ lnηF
M (y′))− f 2
n ψL
(f 2n ;y′
). (5.2.97)
Equating coefficients of ψM gives thus that a solution to Eq. (5.2.92) is given by
(1/ηF
L
)∫dy′(
F(r)
L
(0,y′)− F
(l)L
(0,y′))
×√
ω2εM (y′)µM (y′)− i fn
(∂y′ lnηF
M (y′))− f 2
n ψL
(f 2n ;y′
)
=(
fn,L/ηFM (y)
)∫dy′F(r)
M
(0,y′)
ψL
(f 2n ;y′
). (5.2.98)
Since this holds for all f 2n , it must hold for all f 2. With Eq. (5.2.91a), this gives that
the reflected field is related to the applied field as
∫dy′ F
(l)L
(0,y′)
ψL
(f 2;y′
)
×((
1/ηFL
)√ω2εM (y′)µM (y′)− i f
(∂y′ lnηF
M (y′))− f 2 +
(fL/ηF
M (y)))
=∫
dy′ F(r)
L
(0,y′)
ψL
(f 2;y′
)
×((
1/ηFL
)√ω2εM (y′)µM (y′)− i f
(∂y′ lnηF
M (y′))− f 2 −
(fL/ηF
M (y)))
,
(5.2.99)
5.2 Hybrid mode expansions 91
and the transmitted field is related to the applied field as∫
dy′ F(r)
M
(0,y′)
ψL
(f 2;y′
)
×((
1/ηFL
)√ω2εM (y′)µM (y′)− i f
(∂y′ lnηF
M (y′))− f 2 +
(fL/ηF
M (y)))
=(2/ηF
L
)∫dy′ F
(r)L
(0,y′)
ψL
(f 2;y′
)
×√
ω2εM (y′)µM (y′)− i f(∂y′ lnηF
M (y′))− f 2. (5.2.100)
With piecewise constant response functions (see Fig. 5.5),
ηFM (y) = ηF
j for y j−1 < y ≤ y j, (5.2.101)
Eq. (5.2.99) gives
4
∑j=1
(f j
ηFL
+fL
ηFj
)∫ y j
y j−1
dy′F(l)L
(0,y′)
ψL
(f 2;y′
)
=4
∑j=1
(f j
ηFL
− fL
ηFj
)∫ y j
y j−1
dy′F(r)L
(0,y′)
ψL
(f 2;y′
). (5.2.102)
For the following solution to the homogeneous differential equation, Eq. (5.2.14b),
ψL
(f 2;y′
)= hexp
(i f(y′−1
))+h′ exp
(−i f
(y′−1
)), (5.2.103)
where, in order to fulfil the boundary conditions of Eq. (5.2.22),
h =1
2(h1 − ih2/ f ) , (5.2.104)
h′ =1
2(h1 + ih2/ f ) , (5.2.105)
and after multiplication with (1/h′)exp(i f (y−1)), Eq. (5.2.102) becomes
4
∑j=1
(f j
ηFL
+fL
ηFj
)∫ y j
y j−1
dy′F(l)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
)))
=4
∑j=1
(f j
ηFL
− fL
ηFj
)∫ y j
y j−1
dy′F(r)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
))),
(5.2.106)
92 Scattering from systems that do not display one-to-one coupling of modes
When y0 < y < y1, we divide Eq. (5.2.106) by f1/ηFL + fL/ηF
1 and obtain
∫ y1
y0
dy′F(l)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
)))
+4
∑j=2
f j/ηFL + fL/ηF
j
f1/ηFL + fL/ηF
1
×∫ y j
y j−1
dy′F(l)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
)))
=4
∑j=1
f j/ηFL − fL/ηF
j
f1/ηFL + fL/ηF
1
×∫ y j
y j−1
dy′F(r)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
))). (5.2.107)
Integration of this equation over f from minus to plus infinity gives
F(l)
L1 (0,y) =∫
d f F(l)
L1
(f 2)
exp(i f y) , y0 < y < y1, (5.2.108)
where the integration path lies below all singularities in the complex f -plane and
where
F(l)
L1
(f 2)
= rFL1
(f 2)
F(r)
L1
(f 2), (5.2.109)
F(r)
L1
(f 2)
=1
2π
∫ y1
y0
dy′F(r)L
(0,y′)
exp(−i f y′
), (5.2.110)
with
rFL j =
f j/ηFL − fL/ηF
j
f j/ηFL + fL/ηF
j
. (5.2.111)
When y1 < y < y2, we divide Eq. (5.2.106) by f2/ηFL + fL/ηF
2 and obtain
4
∑j 6=2
f j/ηFL + fL/ηF
j
f2/ηFL + fL/ηF
2
∫ y j
y j−1
dy′F(l)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
)))
+∫ y2
y1
dy′F(l)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
)))
=4
∑j=1
f j/ηFL − fL/ηF
j
f2/ηFL + fL/ηF
2
∫ y j
y j−1
dy′F(r)L
(0,y′)( h
h′exp(i f(y+ y′−2
))+ exp
(i f(y− y′
))).
(5.2.112)
5.2 Hybrid mode expansions 93
Integration of this equation over f from minus to plus infinity now yields for y1 <y < y2 that
F(l)
L2 (0,y)+∫
d ff1/ηF
L + fL/ηF1
f2/ηFL + fL/ηF
2
F(l)
L1
(f 2)
exp(i f y)
=∫
d ff2/ηF
L − fL/ηF2
f2/ηFL + fL/ηF
2
F(r)
L2
(f 2)
exp(i f y)
+∫
d ff1/ηF
L − fL/ηF1
f2/ηFL + fL/ηF
2
F(r)
L1
(f 2)
exp(i f y) , (5.2.113)
where the integration paths lie below all singularities in the complex f -plane and
where
F(r)
L2
(f 2)
=1
2π
∫ y2
y1
dy′F(r)L
(0,y′)
exp(−i f y′
). (5.2.114)
With the use of Eq. (5.2.108), it then follows that
Fr,2 (0,y) =∫
d f Fr,2
(f 2)
exp(i f y) , y1 < y < y2, (5.2.115)
where the integration path lies below all singularities in the complex f -plane and
where
F(l)
L2
(f 2)
= rFL2
(f 2)
F(r)
L2
(f 2), (5.2.116)
F(r)
L2
(f 2)
=1
2π
∫ y2
y1
dy′F(r)L
(0,y′)
exp(−i f y′
). (5.2.117)
Thus we have calculated for every slab j = 1,2,3,4 that
F(l)
L j (0,y) =∫
d f F(l)
L j
(f 2)
exp(i f y) , y j−1 < y < y j, (5.2.118)
where the integration path lies below all singularities in the complex f -plane and
where
F(l)
L j
(f 2)
= rFL j
(f 2)
F(r)
L j
(f 2), (5.2.119)
F(r)
L j
(f 2)
=1
2π
∫ y j
y j−1
dy′F(r)L
(0,y′)
exp(−i f y′
). (5.2.120)
94 Scattering from systems that do not display one-to-one coupling of modes
Analogously, it can be calculated that the transmitted field at the interface x = 0 is
given by
F(r)
M j (0,y) =∫
d f F(r)
M j
(f 2)
exp(i f y) , y j−1 < y < y j, (5.2.121)
where the integration path lies below all singularities in the complex f -plane and
where
F(l)
M j
(f 2)
= tFL j
(f 2)
F(r)
L j
(f 2), (5.2.122)
with
tFL j =
2 f j/ηFL
f j/ηFL + fL/ηF
j
. (5.2.123)
A monochromatic field that propagates under an angle θ with the horizontal axis
generates at the boundary interface x = 0 the field distribution
F(r)
L (ω;0,y) = Reexp [ikLysinθ] , (5.2.124)
Eq. (5.2.120) gives
F(r)
L j
(f 2)
=1
4πi
[exp(i(kL sinθ− f )y j)− exp(i(kL sinθ− f )y j−1)
kL sinθ− f
]
− 1
4πi
[exp(−i(k⋆
L sinθ+ f )y j)− exp(−i(k⋆L sinθ+ f )y j−1)
k⋆L sinθ+ f
].
(5.2.125)
5.3 Numerical results
The analytical results obtained above will now be numerically evaluated for the fol-
lowing geometry. At x < 0, there is a vacuum. At x > 0, we have a dielectric medium
that consists of two horizontal layers in between another dielectric medium. The
dielectric functions ε j(ω), j = 1 · · ·4 for the various layers, read as
ε j(ω) = ε0 +ε0ω2
p j
ω2j −ω2 −2iγ jω
j = 1, . . . ,4. (5.3.1)
The numerical values of the parameters in Eq. (5.3.1) are given in Table 5.3.1. Note
that the first and fourth layer are given identical optical properties. The medium is
5.4 Discussion 95
Slab parameter values
l2 = 200nm l3 = 300nm
ωp1 = 1.5 ωp2 = 4.5 ωp3 = 2.0 ωp4 = 1.5ω1 = 4.0 ω2 = 4.0 ω3 = 2.5 ω4 = 4.0γ1 = 0.10 γ2 = 0.20 γ3 = 0.15 γ4 = 0.10
Table 5.3.1: Slab parameter values, where frequencies are given in units of 1016rad/s.
All media have µ j = µ0.
illuminated by the monochromatic incoming field specified above, where we take
for the angular frequency ω0 = 4.0 · 1015rad/s and for the angle θ = π8. The field
distributions at x = 0 of this incoming field and the resulting transmitted and reflected
fields that were calculated with the above formulae have been plotted in Fig. 5.6.
5.4 Discussion
The usual theory for the solution of boundary value problems is based on the theory of
eigenfunction expansions for separable coordinate systems for the pertinent scalar- or
vectorial wave equations. Then, for geometries of the scattering medium coinciding
with these separable coordinate systems, the scattering problem can be solved in
terms of the appropriate eigenfunctions [125, 126]. If, however, the boundaries of
the scatterer do not coincide fully with a separable coordinate system, this theory
is not applicable but the theory developed above gives the possibility to perform an
exact calculation. An example of such an extension of the usual theory is provided
by the scattering of a wave from the configuration drawn in Fig. 5.2. Moreover, we
would like to observe that scattering by a multilayer slab whose layers are neither
parallel- or perpendicular to the boundary planes of the slab can be exactly calculated
with our theory if a new coordinate system is introduced such that the y-axis stays as
shown in Fig. 5.2, whereas the x-axis is parallel with the layers. Another example of
the extension of the usual theory made possible by our new theory of eigenfunction
expansions is that of the scattering by, for instance, slabs whose optical properties
are changing in both the x and y directions, provided the pertinent wave equations
separate. So, e.g., scattering by a square structure consisting of N layers of constant
refractive index n1 in the x-direction superposed on M layers with constant index of
refraction n2 in the y-direction can be exactly calculated. The theory developed in
96 Scattering from systems that do not display one-to-one coupling of modes
this paper can also be generalized for the analysis of the scattering properties of a
structure consisting of regularly placed pyramids. Such a structure occurs e.g. at
the cornea of moths, [128]. Another example of a now solvable scattering problem
using the methods of this paper would be that of a slab with sinusoidal changing
refractive index n2 (x,y) = Asin(x)+Bsin(y), where A and B denote constants. The
field modes are then the Mathieu functions.
The key feature of the theory, put forward by us, is the introduction of a new set
of modes described in App. C. They can be interpreted as the modes generated by a
driven wire or membrane with a ”driving force” KFM
(f 2;y
). The key property of these
modes is that the completeness relation is in terms of these modes and a related set
of modes of the homogeneous equation, i.e. free space modes, see Eq. (C.1.17). This
enables the expansion of the fields into either set of modes. This essential property
of the modes then leads to an analysis of the continuity conditions quite similar to
those in case of scattering of an incoming wave by a homogeneous medium. The
involved pertinent spectral densities satisfy a linear set of equations. These equations
are very similar compared to the ones obtained in case of scattering by a slab of
homogeneous material, in which case the spectral densities are connected with the
various expansions of the fields into plane waves.
At first sight the numerical implementation of this theory might seem to be rather
difficult. The expansions depend e.g. on the calculation of the roots of a transcen-
dental equation, Eq. (5.2.30), whereafter then several infinite summations have to be
evaluated. However, all these summations are expressed in terms of closed analyt-
ical formulae, viz. the contour integral of Eq. (C.1.16) of App. C, and can thus be
calculated evaluating an integral with explicitly known argument.
5.4 Discussion 97
4.´10-6 6.´10-6 8.´10-6 0.00001y
-1.0
-0.5
0.5
1.0
E
incHΩ0;x=0,yL
2.´10-6 4.´10-6 6.´10-6 8.´10-6 0.00001y
-1.0
-0.5
0.5
1.0
E
trHΩ0;x=0,yL
2.´10-6 4.´10-6 6.´10-6 8.´10-6 0.00001y
-0.2
-0.1
0.1
0.2
E
ref HΩ0;x=0,yL
Figure 5.6: Distributions of the incoming, transmitted and reflected fields at the in-
terface at x = 0 between a homogeneous medium to the left and a horizontal set of
two layers to the right.
Chapter 6
Summary and Outlook
In this chapter, the main conclusions drawn in this thesis are summarized and pos-
sible future research directions are stated. Photonic crystals are manmade structures
that are candidate materials for the realization of an efficient (low loss) and small-
scale control of the flow of light. In this thesis, several theoretical aspects of the
propagation of electromagnetic pulses in one-dimensional photonic crystals are in-
vestigated. Throughout the thesis, the model that is used for the one-dimensional
photonic crystal is the stratified periodic multilayer.
In chapter 2, it is shown, for propagation in a one-dimensional photonic crystal
in which the frequency dispersion and absorption of the slabs are modeled as single
electron resonance Lorentz media, that the wavefront of a generic electromagnetic
pulse propagates at the vacuum speed of light. The wavefront is composed from the
infinite-frequency components of the applied pulse, which are necessarily present in
signals with finite time duration (pulses). The values of all slab refractive indices are
equal to one at infinite frequency because the electrons of the medium are inert. As a
result, the medium inhomogeneity is not ‘felt’ by the wavefront.
The wavefront of the pulse is immediately followed by the first, fastest propagat-
ing, forerunner of the transmitted pulse, the Sommerfeld precursor. This precursor is
built from the very high frequency components of the applied pulse, where very high
frequency means as compared to the electron resonances of the medium. These pulse
components propagate relatively freely as well, as a consequence of the inertia of the
electrons. From a light-ray picture it is concluded that only the light-ray that does
not undergo internal reflections within the photonic crystal before it is transmitted,
100 Summary and Outlook
contributes to the Sommerfeld precursor1. It follows that the Sommerfeld precursor
merely feels the spatial average medium; no effects of the medium inhomogeneity
are found. Since these effects can only be a result of the interference of internally
reflected (but finally transmitted) light-rays, this conclusion is obvious: internally
reflected light-rays were not included in the calculation.
After the Sommerfeld precursor, the second, slightly slower propagating, fore-
runner of the transmitted pulse arrives. This precursor is composed from the very
low-frequency components provided by the applied pulse, where very low means
again as compared to the slab electron resonances. From a temporal sequence of
the multilayer transmittance plots, the dominant low-frequency contributions to the
early transmitted field are seen to be affected by the medium scattering resonances.
The frequency spectrum of the Brillouin precursor, roughly spoken, ‘tunes’ as fol-
lows due to the medium inhomogeneity: positive peaks appear close to the scattering
resonances and a small minimum appears at the Bragg-frequency. The transmitted
Brillouin precursor is calculated semi-analytically as the sum of the individual sta-
tionary phase points.
In chapter 4, the transmission coefficient of the multilayer is derived as the sum of
the transmission coefficients of all individual transmitted light-rays. The basic path
elements for all possible internal reflections are identified as loops, back-and-forth
reflections between interfaces for which the path of the light-ray starts and ends at
the same interface (the path closes in the axial direction, hence the term ‘loop’). With
a derived small set of loop combinatorics rules, it is shown that all possible internal
reflections, and therewith all possible transmitted light-rays, through the multilayer
can be captured in a geometric series, just as in the case of monolayer. From this,
a very simple expression for the transmission coefficient of the multilayer results,
directly in terms of the Fresnel coefficients and exponential propagation factors.
Chapter 5 treats scattering from inhomogeneous boundaries, this situation is in-
evitably encountered in scattering from higher dimensional photonic crystals. The
electromagnetic field modes at both sides of such a boundary do not couple one-to-
one. Generally, one mode on one side of the boundary couples to an infinite number
of modes at the other side. This results in an infinite set of equations for the descrip-
tion of the scattering. However, with the introduction of two different sets of modes,
that satisfy certain completeness relations, the electromagnetic fields at both sides
1At hindsight, a more elegant derivation would proceed by working with the effective homogeneous
index of refraction of the multilayer, since this avoids the inelegant manual selection of the contributing
light-ray.
101
of the boundary can be expanded in the same set of modes. Then, a solution to the
scattering problem is again found from equating the boundary conditions mode one-
by-one. In this description, the mode coupling has become dependent on the position
along the boundary.
Finally, some suggestions for possible future research directions in the field of
theoretic research on photonic crystals are given. For instance, it is relevant to find
out how the group-, signal- and energy velocities, which are quite well-defined con-
cepts for pulse propagation in homogeneous media with frequency dispersion and
absorption, should be defined for pulse propagation in photonic crystals. These quan-
tities all describe various aspects of the traveling pulse, and are therefore relevant for
describing how information is propagated. Since, as is shown in this thesis, realistic
photonic crystals have a complicated transmittance landscape, this will therefore be
a difficult task.
Further, it is useful to find out how the theory of partial coherence of electro-
magnetic waves, that describes the statistical properties of the electromagnetic vector
field, can be extended from homogeneous materials to photonic crystals. As men-
tioned in the introduction, a strong coherence of the light in photonic crystals is
required for several applications, whereas the influence of the statistical properties
of the field on the polarization properties are equally important as well. Since this
coherence is strongly dependent on the dispersion relation, the extension is nontrivial.
For future research topics that lie directly in the smaller line of this thesis, it would
of course be very nice to derive the reflection coefficient of the multilayer from the
light-ray picture as well. Further, there are still many open ends in the theory of
scattering from boundaries between media with mismatching modes. For instance,
how will the photonic band-gap, that results by the virtue of a ‘one-to-many’ coupling
of modes in photonic crystals, appear in a theory that uses the effective one-to-one
mode coupling as in chapter 5?
Appendix A
Accuracy of the calculation of the
Sommerfeld precursor
In this appendix, the accuracy of the Sommerfeld precursor approximation to the field
is estimated. For brevity, a plane wave of a single carrier frequency ωc is consid-
ered, which is perpendicularly incident from the vacuum at x < 0 on a homogeneous
medium at x > 0. The incident pulse has the time dependence E(t) and time duration
T at x = 0. The total electric field in the homogeneous medium can be written as
E(τ,x) =∫
SdωL(ω;x)S(ω)exp
(−i
ξ1(x)
ω− iωτ
), (A.0.1)
L(ω;x) =∞
∑q=0
ω2qc
ω2qexp
(−i
∞
∑r=2
ξr(x)
ωr
), S(ω) =
1
2π
ωcE(ωc)
ω2,
where τ = t − x/c, E(ωc) = 2T
∫ T0 dtE(t)sinωct and
ξr(x) ≡−x
c
1
(r +1)!
dr+1η(ν)
dνr+1|ν=ω−1=0. (A.0.2)
104 Accuracy of the calculation of the Sommerfeld precursor
Here η(ν) = n(ω)|ω=ν−1 and n(ω) is the refractive index of the medium at x > 0,
given by Eq. (2.2.1). Use
(−i)n+1
n!
∫ τ
0dτ′(τ− τ′)nS(τ′,x) =
∫
Sdω
1
ωn+1S(ω)exp
(−iωτ− i
ξ1(x)
ω
), (A.0.3)
S(τ,x)+
(exp
((−i)n+1
n!
∫ τ
0dτ′(τ− τ′)n
)−1
)S(τ′,x)
=∫
Sdωexp
(1/ωn+1
)S(ω)exp
(−iωτ− i
ξ1(x)
ω
), (A.0.4)
to rewrite Eq. (A.0.1) as E(τ,x) = S(τ,x) + (L ∗ S)(τ,x) where convolution is with
respect to the variable τ and L = ε1 + ε2 + ε1 ∗ ε2 with
ε1 ∗S(τ,x) =∞
∑q=1
ω2qc
(−i)2q
(2q−1)!
∫ τ
0dτ′(τ− τ′)2q−1S(τ′,x), (A.0.5)
ε2(x)∗S(τ,x) =
(exp
(−i
∞
∑r=2
ξr(x)(−i)r
(r−1)!
∫ τ
0dτ′(τ− τ′)r−1
)−1
)S(τ′,x).
(A.0.6)
The difference between the actual electric field and the Sommerfeld approximation
is the remainder R = E −S = L∗S. We demand that for some small ε ∈ R,
||R||2 = ||L∗S||2 < ε||S||2, (A.0.7)
where ||R||2 is the L2-norm of the function R on the interval (0,τ),
||R||2 =
√1
τ
∫ τ
0dτ′|R(τ′,x)|2. (A.0.8)
Since L is linear, its norm is (see [129]) ||L|| = sup|| f ||=1 ||L f ||. Therefore ||L f || ≤||L|| and we may require ||L||2 < ε. This requirement is relaxed a little by demanding
||ε1||2 < ε/2 and ||ε2||2 < ε/2 (A.0.9)
and neglecting the ε1 ∗ ε2-term. A typical term in L is of the form
∣∣∣∣∣∣∣∣(−i)n+1
n!
∫ τ
0dτ′(τ′− τ)n
∣∣∣∣∣∣∣∣2
2
=
∣∣∣∣∣∣∣∣(−i
∫ τ
0dτ1
)(−i
∫ τ1
0dτ2
)· · ·(−i
∫ τn
0dτn+1
)∣∣∣∣∣∣∣∣2
2
≤∣∣∣∣∣
∣∣∣∣∣
(−i
∫ τ
0dτ′)n+1
∣∣∣∣∣
∣∣∣∣∣
2
2
= sup|| f ||=1
1
τ
∫ τ
0dτ′(∫ τ′
0dτ′′ f (τ′′)
)2n+2
. (A.0.10)
105
The supremum in the right-hand-side of Eq. (A.0.10) is found by demanding the
functional
I[ f ](τ) =1
τ
∫ τ
0dτ′(∫ τ′
0dτ′′ f (τ′′)
)2n+2
+λ(τ)(1
τ
∫ τ
0dτ′ f (τ′)2 −1
)(A.0.11)
to be stationary under variations in f . Here λ is a Lagrange multiplier. This gives
f = ±1 and ∣∣∣∣∣∣(−i)n+1
n!
∫ τ
0dτ′ (τ′− τ)n
∣∣∣∣∣∣2≤ τn+1
√2n+3
. (A.0.12)
Therefore
||ε1||2 ≤∞
∑q=1
ω2qc τ2q
√4q+1
and ||ε2||2 ≤ exp( ∞
∑r=2
|ξr(x)|τr
√2r +1
)−1. (A.0.13)
This gives∞
∑q=1
ω2qc τ2q
√4q+1
< ε/2,∞
∑r=2
|ξr(x)|τr
√2r +1
< ε/2. (A.0.14)
so if
ωcτ < ε/2 and |ξ2|τ2 =γω2
pax
2cτ2 < ε/2, (A.0.15)
then requirement Eq. (A.0.9) is fulfilled and the Sommerfeld precursor gives an ac-
curate description of the electric field. For the choice ωc = 3.00 ·1015s−1 the inequal-
ity on the left-hand-side of Eq. (A.0.15) gives τ < 0.17ε fs. With γ = 4 · 1013s−1,
ωpa = 2.4 ·1016s−1 and x = 6 ·10−5m the other inequality gives τ < 1.4 ·10−17√
ε s.
So for ε = 0.01 and with oscillation times of ∼ 10−19s (see Fig. 2.5) the approxima-
tion is accurate over ∼ 102 oscillations.
Appendix B
Method of steepest descent
In order to illustrate the method of steepest descent, we use the integral of Eq. (3.4.1),
ER (θ) =∫
dωEL (ω)expΦ(θ;ω) , (B.0.1)
where we have omitted the various N-subscripts and (r)-superscripts, as we will do
throughout this appendix. We assume that EL varies slowly as a function of ω as
compared to Φ in the neighborhood of the relevant stationary points of the latter
function. Let X and Y respectively denote the real and imaginary parts of Φ and
let ξ and η denote respectively the real and imaginary parts of ω. Demanding Φ(1),
which denotes the first-order ω-derivative of Φ, to be independent of the direction
along which the derivative is taken in the complex plane gives the Cauchy-Riemann
equations,
Xξ = Yη, Xη = −Yξ, (B.0.2)
where Xξ = ∂X/∂ξ etcetera. Let τ parametrize the deformed integration path. When
this path follows the steepest slope lines of X , the coordinate derivative must satisfy
(ξ
η
)= ±
(Xξ
Xη
), (B.0.3)
where the dot denotes the τ-derivative and where the plus and minus sign stand for
respectively the steepest ascent and descent lines. From the chain rule for differenti-
ation and from Eqs. (B.0.2) and (B.0.3) it follows that
Y = 0. (B.0.4)
108 Method of steepest descent
This proves that the phase is constant along the lines of steepest descent. The n-th
order stationary points of Φ satisfy
Φ(k) = 0, k = 1, . . . ,n. (B.0.5)
When Φ depends on time θ as well, the stationary points generally sweep out trajec-
tories in ω-space and we write the solutions to Eq. (B.0.5) as
ω = φs (θ) . (B.0.6)
The Taylor expansion of Φ in ω about an n-th order stationary point at ω = φs (θ) is
equal to
Φ(θ;ω) = Φ(θ;φs (θ))+Φ(n+1) (θ;φs (θ))
(n+1)!(ω−φs (θ))n+1 +o
((ω−φs (θ))n+2
).
(B.0.7)
With polar coordinates in the complex ω-plane it easily follows that, at an n-th order
stationary point of Φ, both X and Y have a saddle-point from which n+1 radial lines
of steepest descent and ascent depart. For first-order stationary points, the contribu-
tion to the field can actually be calculated and Fig. B.1 illustrates X at a fixed time θ
close to a first-order stationary point at ω = φs (θ). The angles of the steepest descent
lines of X departing from this point are given by
αs (θ) =1
2
(π− arg Φ(2) (θ;φs (θ))
), (B.0.8)
and the other is at αs + π. When the integration path is taken along the radial line
at angle αs through this point, the parametrization of the path equals ω = φs (θ) +eiαs(θ)τ and the contribution from this stationary point to the field of Eq. (B.0.1) can
be calculated with a quadratic approximation of Φ as
E(s)R (θ) =EL (φs)exp(Φ(θ;φs)+ iαs)
∫dτ exp
(−1
2
∣∣∣Φ(2) (θ;φs)∣∣∣τ2
),
=√
2π EL (φs)exp(Φ(θ;φs)+ iαs)∣∣∣Φ(2) (θ;φs)
∣∣∣− 1
2. (B.0.9)
The symmetry Φ∗N (θN ,ω) = ΦN (θN ,−ω∗) implies that, if Φ has a stationary point
at ω = φs, it also has one at ω = −φ⋆s . When an integration path is used that is
symmetric about the imaginary axis, the contribution from one stationary point equals
the complex conjugate of the other so that both stationary points together contribute
two times the real part of Eq. (B.0.9).
109
ξ
η
X
•
φs (θ) αs (θ)
first-ordersaddle-point
steepestdescent
Figure B.1: Illustration of an instantaneous plot of X (θ;ξ,η) = Re Φ(θ;ω) near a
first-order stationary point of Φ at ω = φs. The steepest descent lines depart from this
point along the radial line at the angle α = αs and in the opposite direction.
Appendix C
Derivation of hybrid completeness
relations
C.1 The special eigenfunction expansions
This section contains the various theorems pertinent to the series expansions of Eqs. (5.2.31)
and (5.2.32), which are fundamental for the theory developed in this paper. Consider
the solutions ψ(1)L
(f 2;y
)and ψ
(2)L
(f 2;y
)of the homogeneous differential equation
(∂2
y + f 2)
ψL
(f 2;y
)= 0, (C.1.1)
that satisfy
ψ(1)L
(f 2;1
)= 1, (C.1.2a)
(∂yψ
(1)L
(f 2;y
))∣∣∣y=1
= 0, (C.1.2b)
ψ(2)L
(f 2;1
)= 0, (C.1.2c)
(∂yψ
(2)L
(f 2;y
))∣∣∣y=1
= 1. (C.1.2d)
The above solutions are independent, because their Wronskian equals one. The gen-
eral solution to Eq. (C.1.1) reads as
ψL
(f 2;y
)= h1ψ
(1)L
(f 2;y
)+h2ψ
(2)L
(f 2;y
), (C.1.3)
112 Derivation of hybrid completeness relations
where h1 and h2 are constants. The solutions to the inhomogeneous differential equa-
tion, (∂2
y + f 2)
ψM
(f 2;y
)= KF
M
(f 2;y
), (C.1.4)
can be constructed with the help of the function
ψ(4)L
(f 2;y,y′
)= ψ
(1)L
(f 2;y
)ψ
(2)L
(f 2;y′
)−ψ
(2)L
(f 2;y
)ψ
(1)L
(f 2;y′
)(C.1.5)
as
ψM
(f 2;y
)=∫ y
0dy′KF
M
(f 2;y′
)ψ
(4)L
(f 2;y′,y
)+ψL
(f 2;y
), (C.1.6)
where
ψL
(f 2;y
)= βψ
(4)L
(f 2;0,y
)−α
(∂y′ψ
(4)L
(f 2;y′,y
))∣∣∣y′=0
(C.1.7)
in which the arbitrary constants α and β fix the boundary conditions of ψM at y = 0
as
ψM
(f 2;y = 0
)= α, (C.1.8a)
(∂yψM
(f 2;y
))∣∣y=0
= β, (C.1.8b)
and where it is supposed that the function KFM is integrable on the interval 0 ≤ y ≤ 1.
It can be verified, that
h1
(∂yψM
(f 2;y
))∣∣y=1
−h2ψM
(f 2;1
)= γ+N
(f 2), (C.1.9)
where γ is a constant and
N(
f 2)
=∫ 1
0dyψL
(f 2;y
)KF
M
(f 2;y
)− γ+βψL
(f 2;0
)−α
(∂yψL
(f 2;y
))∣∣y=0
(C.1.10)
is a discretisation condition for the separation variable f 2. Let G denote the Green
function that satisfies the differential equation
(∂2
y + f 2)
G(
f 2;y,y′)
= δ(y− y′
), (C.1.11)
the boundary conditions
G(
f 2;1,y′)
= h1ρ(
f 2;y′), (C.1.12a)
∂yG(
f 2;y,y′)∣∣
y=1= h2ρ
(f 2;y′
), (C.1.12b)
C.2 Transformation of series 113
with ρ a function to be determined, and the eigenvalue equation
∫ 1
0dy′G
(f 2;y′,y
)KF
M
(f 2;y′
)−ρ(
f 2;y)
γ+βG(
f 2;0,y)
−α(∂y′G
(f 2;y′,y
))∣∣y′=0
= 0. (C.1.13)
From Eqs. (C.1.11) and (C.1.12), it follows that
G(
f 2;y,y′)
=
ψL
(f 2;y
)ρ(
f 2;y′)
if y ≥ y′,
ψL
(f 2;y
)ρ(
f 2;y′)+ψ
(4)L
(f 2;y,y′
)if y ≤ y′.
(C.1.14)
Eq. (C.1.13) then gives
ρ(
f 2;y)
=ψM
(f 2;y
)
N ( f 2). (C.1.15)
From the theory of residues and from Eqs. (C.1.14) and (C.1.15), it follows that
1
2πi
∮
|λ2|=Λ2dλ2 G
(λ2;y,y′
)
λ2 − f 2= G
(f 2;y,y′
)+∑
n
ψL
(f 2n ;y)
ψM
(f 2n ;y′
)
( f 2n − f 2)N′ ( f 2
n ), (C.1.16)
where Λ2 denotes a very large constant, where N′ ( f 2n
)=(dN/d f 2
)∣∣f 2= f 2
nand where
the f 2n are the solutions of the transcendental equation, viz. N
(f 2n
)= 0. In the
limit Λ2 → ∞, the integral in Eq. (C.1.16) vanishes. From applying(∂2
y + f 2)
to
Eq. (C.1.16), one finds the hybrid mode expansion
δ(y− y′
)= ∑
n
ψL
(f 2n ;y)
ψM
(f 2n ;y′
)
N′ ( f 2n )
. (C.1.17)
In the following section, a mode expansion similar to Eq. (C.1.17) will be derived,
but one that only involves the free space functions ψL.
C.2 Transformation of series
Eq. (C.1.17) gives for KFM = 0 that
δ(y− y′
)= ∑
j
ψL
(λ2
j ;y)
ψL
(λ2
j ;y′)
N′0
(λ2
j
) , (C.2.1)
114 Derivation of hybrid completeness relations
where
N0
(f 2)
= −γ+βψL
(f 2;0
)−α
(∂zψL
(f 2;z
))∣∣z=0
, (C.2.2)
and where
λ2j
denote the roots of N0, viz. N0
(λ2
j
)= 0. Eq. (C.2.1) gives a series
expansion of the delta-distribution in the free space modes, but it involves a sum over
different eigenvalues than those of Eq. (C.1.17). In order to obtain an expansion series
for free space with a sum that runs over the eigenvalues f 2n , consider the following.
Suppose that NG, N0 and NK are analytic functions of λ2 in the entire complex plane,
with N0 +NK = N, such that
lim|λ2|→∞
∣∣∣∣∣NK
(λ2)
NG
(λ2)
(λ2 − f 2)N0 (λ2)N (λ2)
∣∣∣∣∣= O(λ−2)
(C.2.3)
where O denotes Landau’s second order symbol. Let, as before,
f 2n
denote the
roots of N, viz. N(
f 2n
)= 0 and
λ2
j
denote the roots of N0, viz. N0
(λ2
j
)= 0.
Then, from the theory of residues, it follows that
1
2πi
∮
|λ2|=Λ2dλ2 NK
(λ2)
NG
(λ2)
(λ2 − f 2)N0 (λ2)N (λ2)=
NK
(f 2)
NG
(f 2)
N0 ( f 2)N ( f 2)
∑j
NG
(λ2
j
)
(λ2
j − f 2)
N′0
(λ2
j
) −∑n
NG
(f 2n
)
( f 2n − f 2)N′ ( f 2
n ). (C.2.4)
In Eq. (C.2.4), the function NG is taken as
NG
(f 2)
= ψL
(f 2;y
)ψL
(f 2;y′
). (C.2.5)
In the limit∣∣Λ2∣∣→ ∞, the integral in Eq. (C.2.4) vanishes on behalf of Eq. (C.2.3)
and, with NG specified as above, this equation gives
0 =NK
(f 2)
N0 ( f 2)
ψL
(f 2;y
)ψL
(f 2;y′
)
N ( f 2)+∑
j
ψL
(λ2
j ;y)
ψL
(λ2
j ;y′)
(λ2
j − f 2)
N′0
(λ2
j
)
−∑n
ψL
(f 2n ;y)
ψL
(f 2n ;y′
)
( f 2n − f 2)N′ ( f 2
n ). (C.2.6)
C.3 Expansion of a plane wave into the free space modes 115
Application of(∂2
y + f 2)
to this equation gives
0 = −∑j
ψL
(λ2
j ;y)
ψL
(λ2
j ;y′)
N′0
(λ2
j
) +∑n
ψL
(f 2n ;y)
ψL
(f 2n ;y′
)
N′ ( f 2n )
. (C.2.7)
With N0 as specified in Eq. (C.2.2), with
NK
(f 2)
=∫ 1
0dyKF
M
(f 2;y
)ψL
(f 2;y
), (C.2.8)
and with the use of Eq. (C.2.1), the result follows as
δ(y− y′
)= ∑
n
ψL
(f 2n ;y)
ψL
(f 2n ;y′
)
N′ ( f 2n )
. (C.2.9)
This is the desired series expansion for the delta-distribution in free space modes,
with a sum that runs over the proper separation constants, that is, the same as those
that appear in the hybrid mode expansion for the delta-distribution.
C.3 Expansion of a plane wave into the free space modes
In this section, we will derive the expansion for the field distribution generated by an
incoming plane wave at a plane boundary. The method used is the standard method
used in interpolation theory for obtaining e.g. the representation of a bandlimited
function in terms of its values at an asymptotically equally spaced set of sampling
points (cardinal series expansion). To this end, we introduce the following contour
integral:
1
2πi
∮
|λ2|=Λ2dλ2 ψL
(λ2;y
)
N (λ2)(λ2 − f 2), (C.3.1)
with
N(
f 2)
=∫ 1
0dyψL
(f 2;y
)KF
M
(f 2;y
)− γ+βψL
(f 2;0
)−α
(∂yψL
(f 2;y
))∣∣y=0
,
(C.3.2)
where KFM =
(k2
L −ω2εMµM
)ψM +
(∂y ln ηF
M
)∂yψM. According to the theorem of
residues,
1
2πi
∮
|λ2|=Λ2dλ2 ψL
(λ2;y
)
N (λ2)(λ2 − f 2)= ∑
n
ψL
(f 2n ;y)
N′ ( f 2n )( f 2
n − f 2)+
ψL
(f 2;y
)
N ( f 2), (C.3.3)
116 Derivation of hybrid completeness relations
where the f 2n are the roots of N
(f 2).
From general theorems concerning the behavior of the solutions of second order
differential equations for large values of the modulus of the separation parameter
f 2 (essentially the well-known WKB approximation), it follows that the asymptotic
behavior of the functions ψL
(f 2;y
)and ψM
(f 2;y
)is equal. From this observation,
it follows that the integral in Eq. (C.3.3) tends to zero in the limit Λ2 → ∞. Thus, one
finds the expansion of the field distribution of an incoming plane wave in terms of the
free space mode functions
ψL
(f 2n ;y)
:
ψL
(f 2;y
)= N
(f 2)∑n
ψL
(f 2n ;y)
N′ ( f 2n )( f 2 − f 2
n ). (C.3.4)
Consider the solution that is obtained when the constants are chosen as h1 = 1 and
h2 = 0, then ψL
(f 2;y
)= ψ
(1)L
(f 2;y
)= cos( f (y−1)).
We observe from Eq. (C.3.4) that
cos( f (y−1)) = N(
f 2)∑n
cos( fn (y−1))
N′ ( f 2n )( f 2 − f 2
n ). (C.3.5)
The Hilbert transform of a function F (x) is defined as [130]
H F(x) =1
πP
∫ +∞
−∞dx′
F (x′)x− x′
, (C.3.6)
where P indicates that the Cauchy principal value of the pertinent integral has to be
taken. Using
H cos( f (y−1))(
f 2)
= sin( f (y−1)) , (C.3.7a)
H
1
f 2 − f 2n
(f 2)
= isgn[Im[
f 2n
]]
f 2 − f 2n
, (C.3.7b)
and taking the Hilbert transform of both sides of Eq. (C.3.5), we end up with the
expansion of the function sin( f (y−1)) into the set of modes cos( fn (y−1)):
sin( f (y−1)) = iN(
f 2)∑n
sgn[Im[
f 2n
]]cos( fn (y−1))
N′ ( f 2n )( f 2 − f 2
n ). (C.3.8)
We observe that the expansions of Eqs. (C.3.5) and (C.3.8) lead to the expansion of
the field distribution of a plane wave,
exp(i f (y−1)) = N(
f 2)∑n
cos( fn (y−1))
N′ ( f 2n )( f 2 − f 2
n )
(1− sgn
[Im[
f 2n
]]). (C.3.9)
Appendix D
List of publications
• [A] R. Uitham and B. J. Hoenders, The Sommerfeld precursor in photonic
crystals, Opt. Comm. 262 (2006).
• [B] R. Uitham and B. J. Hoenders, The electromagnetic Brillouin precursor in
one-dimensional photonic crystals, Accepted for publication in Opt. Comm.
• [C] R. Uitham and B. J. Hoenders, Transmission coefficient of a one-dimensional
layered medium from a light-path sum, JEOS Rapid Publications 3, 08013 (2008)
• [D] B. J. Hoenders, M. Bertolotti and R. Uitham, Set of modes for the de-
scription of wave propagation through slabs with a transverse variation of the
refractive index, J. Opt. Soc. Am. A 24 (2007).
• [E] B. J. Hoenders, M. Bertolotti and R. Uitham, Scattering of waves from a
slab with transverse variation of the refractive index, To be submitted.
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Nederlandse Samenvatting
In dit proefschrift worden verscheidene theoretische aspecten van de propagatie van
electromagnetische pulsen in eendimensionale fotonische kristallen behandeld. In
het laatste hoofdstuk wordt een stap in de richting van tweedimensionale fotonische
kristallen gemaakt. Alvorens het werk en de resultaten te bespreken, worden de be-
langrijkste begrippen in dit proefschrift uitgelegd.
De laatste twintig jaar is er hard gewerkt aan de ontwikkeling van fotonische
kristallen, omdat met deze kunstmatige materialen de voortplanting van electromag-
netische golven kan worden beınvloed. Electromagnetische golven zijn fluctuaties
in het electrische en magnetische veld die zich voortplanten met de snelheid van
het licht. In het spectrum van electromagnetische golven zijn bij achtereenvolgens
toenemende golflengten onder meer de volgende alledaagse voorbeelden te vinden:
rontgenstraling (bekend van onder meer medische beeldvormingstechnieken), zicht-
baar licht en warmtestraling (voelbaar bij straalkachels). De meeste toepassingen
van fotonische kristallen liggen in de manipulatie van electromagnetische golven
met golflengten in en net boven het spectrum van zichtbaar licht. Electromagneti-
sche pulsen dienen als dragers van informatie en bestaan uit een breed spectrum van
verschillende golflengten. Deze gecodeerde signalen kunnen met fotonische kristal-
len worden gestuurd. In het algemeen zal de pulsvorm vervormen onder propagatie
door het fotonisch kristal. Het is dus nuttig om uit te zoeken hoe het fotonisch kristal
een electromagnetische puls beınvloedt.
Sturing van licht met fotonische kristallen heeft ten opzichte van conventione-
lere lichtgeleidingsmethoden zoals reflectie aan metalen spiegels en geleiding door
glasvezelkabels1 een belangrijk voordeel. Ten eerste zijn de energieverliezen bij de
manipulatie van zichtbaar licht met fotonische kristallen veel kleiner dan bij het ge-
1Het natuurkundig principe achter lichtgeleiding in bijvoorbeeld glasvezelkabels is totale interne
reflectie.
128 Nederlandse Samenvatting
bruik van metalen spiegels. Ten tweede kan het licht met fotonische kristallen op de
zeer kleine schaal van de golflengte van het licht zelf worden gestuurd, dit is niet mo-
gelijk bij een sturing door glasvezelkabels. Fotonische kristallen kunnen dus dienen
als een klein en efficient instrument om de propagatie van licht mee te controleren.
De bouwstenen van het fotonisch kristal zijn verschillende dielectrische compo-
nenten. Binnen de eenheidscellen van het kristal worden toegediende electromagne-
tische golven gedeeltelijk gereflecteerd ten gevolge van de contrasten in brekingsin-
dices van de verschillende componenten. Als de afmetingen van de eenheidscellen
nu zodanig zijn, dat de gereflecteerde golven van opeenvolgende eenheidscellen con-
structief interfereren, dan zal het netto resultaat zijn dat het licht sterk wordt gereflec-
teerd aan het kristal. Het interval van golflengten waarbij er geen lopende golf kan
bestaan in een gegeven fotonisch kristal wordt de fotonische “band kloof” genoemd.
Als de band kloof stand houdt voor alle mogelijke voortplantingsrichtingen, dan heet
de band kloof “compleet”. Het tegendeel van reflectie vindt plaats als de gereflec-
teerde golven aan opeenvolgende eenheidscellen van het fotonisch kristal destructief
interfereren, dan wordt de golf als geheel goed doorgelaten. Het fotonisch kristal
bewerkstelligt dus een selectiviteit in de transmissie van electromagnetische golven,
waarbij de selectiecriteria de golflengte en voortplantingsrichting zijn. Naast de toe-
passing van fotonische kristallen als golfgeleider, zijn er al veel meer applicaties be-
dacht, en deze zijn deels zeer succesvol gerealiseerd; hier wordt in de introductie van
het proefschrift uitvoerig op ingegaan. Ook wordt in de inleiding een uitgebreid his-
torisch overzicht gegeven van de ontwikkeling van fotonische kristallen. De rest van
deze samenvatting is gewijd aan het specifieke werk en de bijbehorende resultaten
waarmee dit proefschrift tot stand is gebracht.
In hoofdstuk twee wordt begonnen met het beschrijven van de propagatie van
electromagnetische pulsen door een eendimensionaal fotonisch kristal, de periodie-
ke multilaag2. Om een natuurgetrouw medium te bieden, is ook frequentiedispersie
en absorptie meegenomen, deze zijn steeds gemodelleerd volgens het Lorentz mo-
del met een enkele electronresonantie per laag. De exacte formules voor de ver-
strooiing aan de (periodieke) multilaag zijn bekend in de vorm van sommen van alle
frequentiecomponenten van de toegediende puls vermenigvuldigd met bijbehoren-
de transmissie- of reflectiecoefficienten. Om inzicht te verwerven in de verschil-
lende interessante pulskenmerken, zoals bijvoorbeeld het golffront, of de hieronder
2Er bestaan ook eendimensionale fotonische kristallen met andere geometrieen dan de periodieke
multilaag. Een voorbeeld is een cilinder, waarin de brekingsindex periodiek varieert als functie van de
straal.
129
beschreven precursors, maar ook voor het begrip van groepssnelheid, signaalsnel-
heid en energiesnelheid van de puls, moeten deze exacte uitdrukkingen in meer detail
worden bekeken. Dit komt omdat de verschillende pulskenmerken vaak kunnen wor-
den toegekend aan afzonderlijke bijdragen, zoals die van de hoogfrequente of juist
de laagfrequente componenten. De verschillende, onderscheidbare bijdragen aan het
veld ontstaan als gevolg van de dispersie en absorptie in het medium. De interes-
sante componenten moeten dus uit de exacte uitdrukkingen worden gelicht om het
bijbehorende pulskenmerk zo goed mogelijk te karakteriseren.
In hoofdstuk twee wordt de ‘voorkant’ van de electromagnetische puls die door
een periodieke multilaag is gegaan, onderzocht. Met behulp van wat standaard reken-
technieken volgt direct dat het golffront van de puls zich met de vacuum lichtsnelheid
voortbeweegt en dat het golffront is opgebouwd uit de componenten van de toege-
diende puls met oneindig hoge frequentie. De electronen in het medium zijn inert en
kunnen deze oneindig snelle trillingen van het veld niet volgen. De componenten van
het veld met oneindig hoge frequentie bewegen dus vrijuit, zonder interactie, en zien
het fotonisch kristal als ware het een vacuum.
Onmiddelijk achter het golffront van de electromagnetische puls die door de pe-
riodieke multilaag is gegaan, volgt de zogenaamde Sommerfeld precursor. Dit is de
eerste (snelste) voorloper van de puls, genoemd naar de ontdekker die deze precursor
in 1914 heeft voorspeld voor pulspropagatie in homogene media met frequentiedis-
persie en absorptie. De Sommerfeld precursor is opgebouwd uit de hoogfrequente
componenten van de toegediende puls, die weinig interactie hebben met het medium
(het fotonisch kristal) omdat de electronen de snelle trillingen van deze componenten
van het toegediende veld nauwelijks kunnen volgen. Met de hoogfrequente compo-
nenten van de puls wordt bedoeld: de componenten met frequenties die groter zijn
dan zowel de draagfrequenties van de puls als de electron resonantiefrequenties van
het medium.
De Sommerfeld precursor volgt uit een benaderende uitdrukking voor het door-
gelaten veld die nauwkeurig is in een klein gebied achter het golffront. Er wordt
aangetoond, dat de bijdragen van de lichtstralen, die in het medium reflecties hebben
ondergaan, buiten de benadering vallen en daarom geen bijdrage geven aan de Som-
merfeld precursor. Dit verklaart het gevonden resultaat, dat de Sommerfeld precursor
geen vervorming ondervindt ten gevolge van het contrast in de brekingsindices van
de lagen. Het effect van contrast is immers terug te vinden in de interferentie van ge-
reflecteerde lichtstralen, maar de gereflecteerde lichtstralen worden simpelweg niet
meegenomen. Waar de amplitude van de Sommerfeld precursor in het geval van
130 Nederlandse Samenvatting
een homogeen medium wordt beınvloed door bepaalde parameters van dat medium,
wordt deze in het fotonisch kristal beınvloed door het ruimtelijk gemiddelde van die-
zelfde parameters.
Iets verder achter het golffront van de puls, na de Sommerfeld precursor, volgt de
Brillouin precursor. Deze tweede, iets langzamere, voorloper van de puls is eveneens
genoemd naar haar ontdekker en is ook voorspeld in 1914. De Brillouin precursor
wordt in hoofdstuk drie onderzocht, wederom voor propagatie in een eendimensionaal
fotonisch kristal. De frequentiecomponenten waaruit deze precursor is opgebouwd
zijn goed te herleiden uit in de tijd opeenvolgende plaatjes van de transmittantie van
het kristal. De transmittantie geeft de intensiteit van de doorgelaten puls ten opzichte
van die van de toegediende puls. Gedurende een bepaalde tijd, direct na het tijdsin-
terval waarop de hoogfrequente Sommerfeld precursor het dominante signaal is, is
de transmittantie het grootst voor lage frequenties. De bijdragen van deze laagfre-
quente componenten aan het doorgelaten veld geven de Brillouin precursor. In het
frequentiespectrum van de Brillioun precursor ontstaan pieken ter hoogte van de ver-
strooiingsresonanties van het fotonisch kristal en er ontstaat een minimum ter hoogte
van de band kloof frequenties. Zoals te verwachten is, neemt dit effect van afstem-
ming toe als het contrast in het medium toeneemt.
Kon de berekening van de Sommerfeld precursor nog volledig analytisch gedaan
worden, de Brillouin precursor vergde al een semi-analytische aanpak; de dominante
bijdragen werden immers op het oog bepaald uit numeriek verkregen plaatjes van de
transmittantie. Aan de hand van diezelfde plaatjes valt te verwachten, dat nog verder
achter het golffront zelfs een semi-analytische berekening erg gecompliceerd wordt.
Dit komt namelijk omdat er dan erg veel verschillende frequentiecomponenten zijn
die allemaal ongeveer evenveel bijdragen aan het signaal. Voor een periodieke mul-
tilaag met slechts vijf lagen met elk diktes van enkele honderden nanometers zijn er,
voor realistische waarden van de overige medium parameters, al meer dan honderd
van zulke bijdragen.
Omdat het werkingsprincipe van een fotonisch kristal toch heel simpel is, name-
lijk periodieke reflectie, moet dit ergens tot uiting komen in de uitdrukkingen voor de
transmissie- en reflectiecoefficient van de multilaag. Bij de gebruikelijke afleiding3
van deze coefficienten zijn de elementaire processen niet eenvoudig te herleiden in de
resulterende uitdrukkingen. Deze elementaire processen zijn transmissie en reflectie
aan interfaces, welke resulteren in de bijbehorende Fresnel coefficienten in de ampli-
3De transmissie- en reflectiecoefficienten worden gewoonlijk afgeleid met behulp van de transfer-
matrix methode.
131
tude van het veld, en propagatie in de lagen, hetgeen resulteert in een exponentiele
‘propagatiefactor’. In hoofdstuk vier wordt de transmissiecoefficient van de multi-
laag opnieuw afgeleid, maar nu als som van de amplitude coefficienten die behoren
bij alle mogelijke paden waarlangs het licht door het fotonisch kristal kan gaan. Een
afleiding op deze manier leidt tot een eenvoudige uitdrukking voor de transmissieco-
efficient van de multilaag.
In hoofdstuk vijf wordt gekeken naar verstrooiing van electromagnetische golven
aan media, waarvan de brekingsindex varieert langs het vlak van inval. Bij meer-
dimensionale fotonische kristallen, waarin de brekingsindex periodiek varieert langs
meerdere onafhankelijke richtingen, doet deze situatie zich altijd voor. Het reken-
technische verschil tussen verstrooiing aan homogene randen en aan inhomogene
randen is, dat de zogenaamde modes van het electromagnetische veld aan weerszij-
den van de rand in het eerste geval wel en in het laatste geval niet een-op-een koppe-
len. De modes van het electromagnetische veld in een bepaald gebied (dus binnen of
buiten het medium) zijn gedefinieerd als de oplossingen van de vergelijkingen waar-
aan de golven moeten voldoen, dus de vergelijkingen die gelden in dat gebied. Als
de modes binnen- en buiten het medium niet een-op-een koppelen, kunnen de velden
aan de rand niet per mode worden vergeleken en is berekening van de verstrooiing
niet triviaal.
Echter, er wordt een relatie afgeleid waarin zowel de modes van het electromag-
netische veld binnen- als buiten het medium voorkomen. Middels een transformatie
is deze formule om te schrijven naar een gelijksoortige relatie die enkel de modes
van het veld buiten het medium bevat. Omdat met beide relaties de velden in- en
buiten het medium in een en dezelfde set van modes zijn te ontwikkelen, kunnen de
velden aan de rand alsnog per mode worden vergeleken. Dit levert oplosbare alge-
braısche vergelijkingen op voor de modes van de verstrooide velden aan de rand van
het medium. Men vindt exact dezefde uitdrukkingen als de Fresnel transmissie- en
reflectiecoefficienten voor verstrooiing aan homogene interfaces, waarin nu echter de
medium parameters van de positie langs de rand afhangen. De stap naar tweedimensi-
onale, rechthoekige, kristallen kan worden gemaakt door opeenvolgende inhomogene
lagen te scheiden door een homogene laag. Op elk van de zo ontstane paren van na-
burige homogene en inhomogene lagen kan bovenstaande theorie worden toegepast.
Vervolgens kan de dikte van de homogene lagen gelijk aan nul worden gesteld om de
verstrooiing aan het originele systeem te verkrijgen.
Acknowledgements
Allereerst bedank ik mijn dagelijks begeleider Bernhard Hoenders voor de zeer pret-
tige samenwerking. Je had veel geduld, vertrouwen en was altijd optimistisch. Met
vele jaren onderzoekservaring en een brede blik herkende je veel van mijn proble-
men en hielp je of verwees je me naar de juiste literatuur. Ook spreek ik hier mijn
dank uit voor de vele interessante reizen die ik mocht maken in de afgelopen jaren
naar zomerscholen, conferenties en dergelijke. Met name de periode in Rome was
bijzonder aangenaam.
Mijn promotor Jasper Knoester ben ik erkentelijk voor zijn sturing tijdens kri-
tische momenten gedurende mijn promotie. Ondanks dat het me niet lukte om me
aan alle tijdsplanningen te houden, en ondanks dat bepaalde resultaten misschien wat
tegenvielen, bleef je toch steeds geduldig. Dat vond ik erg prettig. Dat ik, na afloop
van mijn contract, nog van mijn werkplek gebruik kon maken, heeft mij ook zeer
geholpen in de afronding van de promotie.
I am grateful to the members of reading committee Hans De Raedt, Paul Urbach
and Ari Friberg for spending their valuable time to read my thesis carefully.
De vele collega’s die kwamen en deels gingen gedurende de afgelopen jaren
bedank ik allen voor hun behulpzaamheid en de prettige sfeer op het instituut. Ik
noem hier Cristi Marocico, Arend Dijkstra, Thomas la Cour Jansen, Dirk Jan Heijs,
Bas Vlaming, Joost Klugkist, Andrea Scaramucci, Chungwen Liang, Santanu Roy,
Sergei Artyukhin, Wissam Chemissany, Dennis Westra, Martijn Eenink, Diederik
Roest en Hendrikjan Schaap. Wijnand Broer, die mij misschien gaat ‘opvolgen’ als
promovendus van Bernhard, wens ik veel succes en een goede tijd de komende jaren.
De secretaresses Ynske, Iris, Annelies en Sietske bedank ik voor de administratieve
hulp.
Omdat vrije tijd het broodnodige complement van het werk vormt, wijd ik hier
ook nog een alinea aan. Buiten de universiteit hield ik mij de afgelopen jaren onder
134 Acknowledgements
meer bezig met muziek, volleybal en skaten. Wekelijkse pret was de volleybaltraining
op de woensdagavond met aansluitend urenlange evaluatiesessies. Daarnaast was het
heerlijk om de weken ’s zomers af te sluiten met de vrijdagavondskatetochten. Voor
de leuke tijd naast het werk en/of de trouwe vriendschap bedank ik Frank, Paul,
Nikos, Benny, Annemarie, Floris, Hans en Truuske en Frits.
Tenslotte bedank ik mijn ouders, mijn zus en de rest van de familie voor hun
steun.