University of Groningen Electromagnetic pulse propagation

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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


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)


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


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)


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)


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)


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-


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


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)


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)


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.


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)


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.


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-


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


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.


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


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)


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′


, (4.3.17a)

tqq′ =2µ−1

q

√εqµq − εLµL sin2 θL

µ−1q

√εqµq − εLµL sin2 θL +µ−1

q′


, (4.3.17b)


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-


ficients take on the expressions

rqq′ =ε−1

q′

√εq′µq′ − εLµL sin2 θL − ε−1

q


ε−1q

√εqµq − εLµL sin2 θL + ε−1

q′


, (4.3.22a)

tqq′ =

√εqµq′

εq′µq

2ε−1q


ε−1q

√εqµq − εLµL sin2 θL + ε−1

q′


, (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


∈ 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)


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.


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.


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)


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


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)


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)


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)


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)


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)


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.


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)

√∂2

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)

√∂2

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)


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.


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)


5.2.1 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


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)


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.


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


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)


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)


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)


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,


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)


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

√∂2

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)


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


))+ exp

(i f(y− y′

))),

(5.2.106)


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


))+ 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


))+ 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


))+ 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


))+ exp

(i f(y− y′

)))

+∫ y2

y1

dy′F(l)L

(0,y′)( h


))+ 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


))+ exp

(i f(y− y′

))).

(5.2.112)


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

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

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)


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

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


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)


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)


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.

Documents

University of Groningen Electromagnetic pulse propagation