# The Sommerfeld precursor in photonic crystals

**ABSTRACT** We calculate the Sommerfeld precursor that results after transmission of a generic electromagnetic plane wave pulse with transverse electric polarization, through a one-dimensional rectangular N-layer photonic crystal with two slabs per layer. The shape of this precursor equals the shape of the precursor that would result from transmission through a homogeneous medium. However, amplitude and period of the precursor are now influenced by the spatial average of the plasma frequency squared instead of the plasma frequency squared for the homogeneous case.

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**ABSTRACT:**We investigate the nonlinear propagation of few-cycle rectangular laser pulses on resonant intersubband transitions in semiconductor quantum wells using an iterative predictor–corrector finite-difference time-domain method. An initial 2π rectangular pulse will split into Sommerfeld–Brillouin precursors and a self-induced transparency soliton during the course of propagation. The duration of generated soliton depends on the carrier-envelope phase of the incident pulse. In our case, not only the near-resonant frequency components but also the low frequency components could contribute to the generation of the soliton pulse when the condition of multi-photon resonance is satisfied. The phase-sensitive property of the solitons results from the phase-dependent distribution of high and low frequency sidebands of few-cycle rectangular pulses.Optics Communications 08/2011; 284(s 16–17):4059–4063. · 1.54 Impact Factor - SourceAvailable from: Bernhard J Hoenders[Show abstract] [Hide abstract]

**ABSTRACT:**We have calculated the electromagnetic Brillouin precursor that arises in a one-dimensional photonic crystal that consists of two homogeneous slabs which each have a single electron resonance. This forerunner is compared with the Brillouin precursor that arises in a homogeneous double-electron resonance medium. In both types of medium, the precursor consists of the components of the applied pulse that have their frequencies below the lowest of the two electron resonances. In the inhomogeneous medium however, the slab contrast starts affecting the precursor field after a certain rise time of the precursor: its spectrum starts to peak at the geometric scattering resonances of the medium whereas minima appear at the Bragg-scattering frequencies.Optics Communications 12/2008; · 1.54 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The large amount of light emitted from a light emitting diode (LED) being trapped inside the semiconductor structure is the consequence of the large value of the refractive index. The total internal reflection (light incident on a planar semiconductor/air interface is totally internally reflected if the angle of incidence exceeds the critical value determined by Snell’s law) is the major factor responsible for the small light extraction efficiency (other important contributions to the losses are the internal absorption and blocking of the light by contacts). The typical LED structure comprising a number of layers most of which have high refractive index could be considered as a multilayer waveguide that could support a large number of trapped guided modes. The paper reviews approaches to enhanced light extraction grouped into two sets depending on whether their application results in the change in the spontaneous emission (either the spontaneous emission rate or the angular distribution, or both): (1) molding of the flow of light emitted from the active region by the modification of the chip shape or the surface morphology to increase the light intensity; and (2) modification of spontaneous emission, for example, by placing of the light emitting region inside the optical cavity. Special attention is given to LEDs made from nitrides of elements of group III (InAlGaN) that cover a large part of visible and ultraviolet (UV) spectra and are considered as a major candidate for sources for the solid-state general illumination. An Appendix contains review of numerical models used to study the light extraction.Physics Reports 02/2011; 498(4):189-241. · 22.91 Impact Factor

Page 1

The Sommerfeld precursor in photonic crystals

R. Uitham*, B.J. Hoenders

University of Groningen, Institute for Theoretical Physics and Materials Science Center, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands

Received 28 September 2005; received in revised form 9 December 2005; accepted 28 December 2005

Abstract

We calculate the Sommerfeld precursor that results after transmission of a generic electromagnetic plane wave pulse with transverse

electric polarization, through a one-dimensional rectangular N-layer photonic crystal with two slabs per layer. The shape of this precur-

sor equals the shape of the precursor that would result from transmission through a homogeneous medium. However, amplitude and

period of the precursor are now influenced by the spatial average of the plasma frequency squared instead of the plasma frequency

squared for the homogeneous case.

? 2006 Elsevier B.V. All rights reserved.

PACS: 42.25.Bs; 42.70.Qs; 78.20.Ci

Keywords: Photonic crystal; Sommerfeld precursor; Pulse propagation

1. Introduction

The propagation of electromagnetic pulses in photonic

crystals [1] exhibits many interesting phenomena. Most

familiar is the photonic band-gap. This effect allows pho-

tonic crystals to be applied in for instance information

technology as small-scale and low-loss wave-guides [1], or

in fundamental research as devices that control spontane-

ous atomic photon emission [2]. Another interesting effect

of a photonic crystal is that it can reduce the magnitude

of the group velocity of an electromagnetic pulse consider-

ably [3]. Theory predicts that this group speed can approach

zero in photonic structures with many periods [4]. This

allows for applications of photonic crystals as optical delay

lines or as data storage compounds [5]. Not only small

group velocities have been observed in photonic crystals,

also superluminal group velocities and photon tunnelling

have been measured [6–9].

The theory for pulse propagation in homogeneous

dielectric media is extensively treated in [10–12]. Due to dis-

persion and absorption in the guiding medium, a pulse sep-

arates into distinct parts [11,12] in configuration space. Its

wavefront propagates at the vacuum speed of light. Imme-

diately behind it, the Sommerfeld precursor [11,12] emerges.

The amplitude of this precursor is very small as compared

to the applied pulse and its period is of the order of

10?19s, both depending on material constants, propagation

distance and time. Behind this precursor, the ultraviolet

Brillouin precursor [11,12] emerges. Both precursors have

been observed for microwaves [13] transmitted through

guiding structures that have dispersion characteristics simi-

lar to those for dielectrics and for optical pulses [14,15] in

water and in GaAs. Behind the Brillouin precursor, the per-

iod of the transmitted signal tunes to the applied pulse per-

iod which marks the transmission of the main part of the

pulse.

Instead of the exponential amplitude decay with propa-

gation distance for the main part of the pulse, its Sommer-

feld precursor amplitude decreases roughly with the inverse

square root of propagation distance [11,12]. This long

range persistence may allow for applications of precursors

to underwater communication or medical imaging [16]. In

this paper, the Sommerfeld precursor is calculated after

transmission through an N-layer medium, serving as a pro-

totype of an inhomogeneous medium.

0030-4018/$ - see front matter ? 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2005.12.077

*Corresponding author. Tel.: +31 0 503634958.

E-mail address: r.uitham@rug.nl (R. Uitham).

www.elsevier.com/locate/optcom

Optics Communications 262 (2006) 211–219

Page 2

This paper has been organized as follows. In Section 2

the N-layer medium is modelled. Section 3 is devoted to

the applied pulse. A brief review of the plane wave transmis-

sion coefficient for the N-layer medium is given in Section 4.

In Section 5, we will determine the wavefront of the trans-

mitted pulse and in Section 6 the transmitted pulse immedi-

ately behind the wavefront, the Sommerfeld precursor, is

investigated. The influence of the layered inhomogeneous

medium structure onto this precursor is discussed in Section

7. We conclude in Section 8. In the appendix, the accuracy

of the Sommerfeld precursor description is estimated.

2. Model for the N-layer medium

In Fig. 1 our model for the rectangular one-dimensional

N-layer medium is depicted and in this section the spatial

dependence of its refractive index is given. Each of the lay-

ers k = 1,...,N contains two homogeneous dielectric slabs

of widths a and b such that

a þ b ¼ K.

These slabs have refractive indices naand nbrespectively.

The N-layer medium serves as a reference homogeneous

medium when nb= naor when b = 0. It serves as a pho-

tonic crystal when nb5 naand at the same time b 5 0. It

is placed in a surrounding homogeneous medium with

refractive index ns. The media are infinitely extended in

the yz-plane. A medium with refractive index nj(j = a, b,

s) is referred to as medium j.

Now the frequency dependence of the refractive indices

of the media will be characterized. The atoms of medium j

have mjelectronic resonances and frjand crjdenote the

oscillator strength and absorption parameter correspond-

ing to the rth resonance at angular frequency xrj. The oscil-

lator strengths satisfy the Thomas–Reiche–Kuhn sum rule

[17, p. 261],

ð1Þ

X

mj

r¼1

frj¼ 1;

ð2Þ

to be used in Section 6. The plasma angular frequency of

medium j is xpj. For the dependence of the refractive index

on the angular frequency x of the electromagnetic field, the

Lorentz model for atomic polarization and the Clausius–

Mosotti relation [17, p. 266] have been used to give

njðxÞ ¼

1 þ2

3

X

mj

r¼1

frjx2

pj

x2

rj? x2? icrjx

frjx2

x2

!1=2

?

1 ?1

3

X

mj

r¼1

pj

rj? x2? icrjx

!?1=2

.

ð3Þ

In Fig. 2 the frequency dependence of the real and imag-

inary parts of a typical refractive index neof the form of

Eq. (3) is illustrated for a homogeneous example medium

e with 10 electron resonances (me= 10). The used values

for neare a plasma frequency xpe= 2.4 · 1016s?1and ten

electron resonances at xre= r · 1016s?1with all oscillator

strengths equal, fre= 0.1 and absorption parameters cre=

(20 + r) · 1014s?1. As well, the first order expansion of ne

about infinite frequency,

n1

eðxÞ ¼ 1 ?x2

hasbeenplotted,sincethisapproximationwillbeusedinthe

Sommerfeld precursor theory. Before proceeding to the pre-

cursor, the applied pulse is specified in the following section.

pe

2x2

ð4Þ

3. Applied pulse

The rectangular pattern in the variation of the optical

properties throughout space, as imposed by the N-layer

medium described in the previous section naturally favors

the use of a cartesian coordinate system. The entrance

plane defines the origin of the x-axis and the other two

coordinates are chosen such that a right-handed coordinate

system is obtained.

The electromagnetic fields propagate in the xy-plane

into the directions of the arrows (see Fig. 1) and have trans-

verse polarization such that Eð0Þ

propagate forwards and primed fields propagate back-

wards with respect to the x-axis. The N-layer medium is

irradiated from the left with a one-dimensional plane wave

packet Es0. The applied pulse is specified such that it prop-

agates under an angle hswith the x-axis. The propagation

jk¼ Eð0Þ

jk^ z. Unprimed fields

Fig. 1. Model for a one-dimensional photonic crystal with electric fields.

Fig. 2. Real and imaginary parts of the example refractive index. The

dotted line is its first order expansion of neabout infinite frequency.

212

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

Page 3

axis of the pulse in medium s is therefore related to the xy-

coordinate system as

rs¼ xcoshsþ y sinhs.

The propagation angles in other media for a wave com-

ponent at frequency x can be derived from Fresnel’s

equation,

ð5Þ

njðxÞsinhjðxÞ ¼ nsðxÞsinhs.

Decompose the time-dependence of the fields in the com-

plete basis of exponentials,

Z

where

Z

All time-harmonic components satisfy the wave-equation,

which reads in a cartesian coordinate system as

ð6Þ

Eð0Þ

jkðt;x;yÞ ¼

dx~Eð0Þ

jkðx;x;yÞe?ixt;

ð7Þ

~Eð0Þ

jkðx;x;yÞ ¼1

2p

dtEð0Þ

jkðt;x;yÞeixt.

ð8Þ

njðxÞ2

c2

o2

ot2?o2

ox2?o2

oy2

!

~Eð0Þ

jkðx;x;yÞe?ixt¼ 0;

ð9Þ

where c is the speed of light in vacuum. For the applied

field Es0, we allow only for plane wave solutions that prop-

agate in the above specified direction. This gives

Z

where

Z

is the spectral density of the applied field at the line rs= 0,

see Fig. 3. Further,

ffiffiffiffiffiffiffiffiffiffiffiffi

where kj¼ ðkj

ky¼x

x2

c2njðxÞ2?k2

c

Es0ðt;rsÞ ¼

dx~Es0ðxÞe?ixtþiksðxÞrs;

ð10Þ

~Es0ðxÞ ¼1

2p

dtEs0ðt;rs¼ 0Þeixt

ð11Þ

kj¼

kj? kj

p

;

ð12Þ

x;ky;0Þ with

cnsðxÞsinhs;

?

kj

x¼

y

?1=2

¼x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

njðxÞ2?nsðxÞ2sin2hs

q

.

ð13Þ

Now the spectral density of Eq. (11) will be calculated

for a generic pulse of finite time duration. The applied field

is constructed such that at rs= 0, it is zero at times

t 62 [0,T] with T > 0 finite,

Es0ðt;rs¼ 0Þ ¼ Es0ðtÞI½0;T?ðtÞ.

Here Es0ð0Þ ¼ Es0ðTÞ ¼ 0 and

1for

t 2 ½0;T?;

0for

t 62 ½0;T?.

A convenient basis for Es0ðtÞ is a trigonometric one,

X

~Es0ðxmÞ ¼2

T

0

ð14Þ

I½0;T?ðtÞ ¼

?

ð15Þ

Es0ðtÞ ¼

1

m¼0

~Es0ðxmÞsinxmt

ZT

dtEs0ðtÞsinxmt.

ð16Þ

Here

xm¼mp

in order for Es0ðtÞ to vanish at t = 0,T. The coefficients

~Es0ðxmÞ are the spectral weights of the unmodulated infi-

nitely long lasting T-periodic signal Es0ðtÞ at its carrier

angular frequencies xm. Only pulses with finite carrier fre-

quencies are considered,

T

ð17Þ

~Es0ðxmÞ ¼ 0

where M is a nonnegative integer. With Eq. (16), the spec-

tral density Eq. (11) of the applied field at rs= 0 becomes

X

Eq. (10) with (19) together describe a generic plane wave

pulse of finite time duration that propagates under an angle

hs with the x-axis in the surrounding medium. For an

applied plane wave packet Es0, the transmitted field EsN

is obtained from it via the plane wave transmission coeffi-

cient for the N-layer medium, which is calculated in the

next section.

for

m > M;

ð18Þ

~Es0ðxÞ ¼1

2p

m

xm~Es0ðxmÞ

x2? x2

m

ð?1ÞmeixT? 1

??.

ð19Þ

4. N-layer medium plane-wave transmission coefficient

For plane wave transverse polarized fields, the temporal

Fourier coefficients of the electric fields in homogeneous

media a surrounding a vertical homogeneous slab b, the

fields being evaluated at a mutual horizontal distance K,

are related via the corresponding unimodular transfer

matrix M [18,19],

~E0

ak?1

The transfer matrix is constructed from propagation and

dynamical matrices,

~Eak

~E0

ak

!

¼ M

~Eak?1

!

¼

A

B

D

C

??

~Eak?1

~E0

ak?1

!

.

ð20Þ

Pj¼

eikj

0

xj

0

e?ikj

xj

!

;

Dj¼

11

ckj

x=x

?ckj

x=x

??

;

ð21Þ

Fig. 3. The line rs= 0 is the wavefront at t = 0 and the applied field

propagates along the rs-axis.

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

213

Page 4

as M ¼ PaD?1

here for transverse electric polarization. This gives the en-

tries of M as

?ka

kb

x

?ka

kb

x

kb

x

ka

x

aDbPbD?1

bDa. The latter in Eq. (21) is given

A ¼ eika

xa

coskb

xb þi

2

kb

ka

x

x

þka

x

kb

x

!

sinkb

xb

!

;

B ¼i

2eika

xa kb

x

ka

x

x

!

sinkb

xb;

C ¼?i

2e?ika

xa kb

x

ka

x

x

!

sinkb

xb;

D ¼ e?ika

xa

coskb

xb ?i

2

þka

x

kb

x

!

sinkb

xb

!

.

ð22Þ

The fields at both sides from the N-layer medium in a sur-

rounding homogeneous medium a are related via

CN DN

~E0

a0

~EaN

~E0

aN

!

¼

AN

BN

??

~Ea0

!

;

AN

CN DN

BN

??

?

A B

C D

??N

ð23Þ

.

The entries AN,...,DNcan be found by solving Eq. (20)

with the discrete Mellin transform method [18]. One finds

AN¼ AaN? a?N

BN¼ BaN? a?N

DN¼ DaN? a?N

a ? a?1?aN?1? a?Nþ1

a ? a?1;

a ? a?1?aN?1? a?Nþ1

a ? a?1

CN¼ CaN? a?N

;

a ? a?1;

.

a ? a?1

q

ð24Þ

Here a?1¼1

of the transfer matrix. The correction of the surrounding

medium a to a medium s gives

CN

DN

2ðA þ DÞ ?1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðA þ DÞ2? 4are the eigenvalues

~EsN

~E0

sN

!

¼ D?1

sDa

AN

BN

??

D?1

aDs

~Es0

~E0

s0

!

ð25Þ

and the transmission coefficient of the N-layer medium fol-

lows as (f ?ka

tN?

~Es0

sN¼0

?4f

ðf ? 1Þ2AN? ðf2? 1ÞðCN? BNÞ ? ðf þ 1Þ2DN

Via the transmission of the applied pulse of Section (3)

through the N-layer medium, we arrive at the transmitted

pulse EsN. In the following section, we determine the wave-

front and in the section thereafter, we investigate its shape

immediately behind the wave front.

x

x)

ks

~EsN

??????~E0

¼

.

ð26Þ

5. Wavefront of the transmitted signal

In order to describe the behavior of the transmitted

pulse immediately behind the wavefront, we will first deter-

mine the wavefront itself. Hereto reconsider the expression

for the applied pulse, Eq. (10). Its integrand is analytic at

the real axis and in the upper half frequency plane. Hence

the integration path may freely be deformed from the real

axis into path S, see Fig. 4. This path follows the real axis

up to a semicircle detour in the upper half plane with center

x = 0 and radius X. With S as integration path, traversed

from x = ? 1 to +1, the transmitted pulse follows from

Eqs. (10) and (26) as

Z

where the factor e?iks

x = NK.

From Eq. (3) it follows that nj(jxj = 1) = 1. Hence

spectral components at infinite frequencies experience the

N-layer medium as if it were a vacuum. So, of all frequency

components, these are the fastest propagating components.

When present in the pulse, solely these infinite frequency

components will form the wavefront at positions ahead

of the starting line rs= 0, i.e. after propagation.

The contribution from these fast propagating frequency

components to the complete field, Eq. (27), can be isolated

by choosing the integration path radius X very large with

respect to the frequency parameters of the refractive

indices,

EsNðt;rsÞ ¼

S

dxtNðxÞe?iks

xðxÞNK~Es0ðxÞe?ixtþiksðxÞrs;

ð27Þ

xðxÞNKmatches the phase of the field at

X?1maxðxpj;xj;cjÞ ¼ ?;

where ? is a small positive real number: 0 < ? ? 1. The ra-

dius X is kept finite in order for the integral in Eq. (27) to

be well-defined for all values of (t,rs). When using this

large radius X, the refractive indices, when evaluated on

S, are approximated accurately by their zeroth order

expansions about infinite frequency, since

ð28Þ

njðxÞjx2S¼ 1 þ Oð?2Þ.

On S, the transmission coefficient for the N-layer system is

within this zeroth order approximation equal to

tNðxÞjx2S’ tNðxÞjnj¼1¼ eix

Denote the fastest travelling part (or very high frequency

part) of the transmitted field EsNof Eq. (27) as E0

expansion of nj about infinite frequency corresponds to

ð29Þ

cNKcoshs.

ð30Þ

sN. The

Fig. 4. The path S together with all frequency parameters in the complex

frequency plane.

214

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

Page 5

an expansion of the transmitted field about E0

equals

Z

¼

S

sN, which

E0

sNðt;rsÞ ¼

S

dxtNðxÞe?iks

xðxÞNK~Es0ðxÞe?ixtþiksðxÞrsjnj¼1

Z

dx~Es0ðxÞe?ixsðt;rsÞ;

ð31Þ

where the real function

sðt;rsÞ ? t ? rs=c.

When we explicate the spectral density in Eq. (31),

Z

? ð?1Þme?ixðsðt;rsÞ?TÞ? e?ixsðt;rsÞ

it is clear that both exponents have arbitrary large negative

real parts (for an arbitrary large radius X) when s < 0. The

contribution from the part of the integration near and at

the real axis is arbitrary small as well since the spectral den-

sity~Es0ðxÞ behaves as

integration path for the first term in Eq. (33) can still be ta-

ken as S in order to have exponential decay with X. For the

second term in Eq. (33), the path should be deformed into a

very large semicircle in the lower half frequency plane, but

then one encounters the branch lines from the refractive in-

dex and the poles at x = ±xm (although the complete

expression Eq. (33) has no poles at these positions, the sep-

arate terms do), giving a nonzero signal for s > 0. Hence

the wavefront is given by the equation s = 0. The quantity

s is therefore the time elapse after the wave-front has

passed. Now that the location of the wavefront has been

determined, we can proceed to the transmitted pulse imme-

diately behind the wavefront.

ð32Þ

E0

sNðt;rsÞ ¼

S

dx

X

m

xm~Es0ðxmÞ

x2? x2

m

??;

ð33Þ

1

x2for jxj large. When 0 < s < T, the

6. Sommerfeld precursor

The Sommerfeld precursor starts immediately behind

the wavefront. Since the wavefront serves as a reference

line, it is convenient to use the set of independent coordi-

nates (s,rs) instead of the set (t,rs). In terms of these, the

complete transmitted field, Eq. (27), reads as

Z

Here S is still the integration path with a semicircle in the

upper half plane, see Fig. 4. The zeroth order expansions

about infinite frequency, were used to determine the wave-

front. In order to derive the Sommerfeld precursor, their

first order expansions will be used,

EsNðs;rsÞ ¼

S

dxtNe?ix

cks

xNKeiðks?x

cÞrse?ixs~Es0.

ð34Þ

njðxÞ ¼ 1 ?x2

The refractive indices are contained in the expression for

the transmitted wave, Eq. (34), through the factors

tNe?iks

andeiðks

pj

2x2.

ð35Þ

xNKðaÞ

x?x

cÞrsðbÞð36Þ

and the high frequency behavior of these factors will be

determined in the following.

With Eq. (35), the high frequency behavior of Eq. (36b)

immediately follows as

eiðks

x?x

cÞrs??

nsðxÞ¼1?

x2

2x2

ps

¼ e?i

x2

psrs

2c

1

x.

ð37Þ

For the manipulation of Eq. (36a), it is convenient to con-

sider another expanded form for tN. Let rjj0 denote the single

boundary reflection factor for an electric field that propa-

gates in medium j and is reflected at a boundary with med-

ium j0and let tjj0 denote the corresponding single boundary

transmission factor. For transverse electric polarization,

rjj0 ¼kj

x? kj0

kj

x

xþ kj0

x

;

tjj0 ¼ 1 þ rjj0;

ð38Þ

see [20]. The x-component of the propagation factor over a

distance j through medium j is

pj¼ eikj

The slab a-transfer matrix elements of Eq. (22) can be

expressed in these factors,

xj.

ð39Þ

A ¼ tabtbapaðpb? r2

B ¼ rbat?1

C ¼ rabt?1

D ¼ t?1

and we can choose to expand the transmission coefficient

tN in powers of p, which is a propagation path-length

ordering of the wave within the N-layer medium. The first

few terms of this path-length expansion of tNare

abp?1

bÞ;

bÞ;

abt?1

abt?1

abt?1

bapaðpb? p?1

bap?1

bap?1

aðpb? p?1

aðp?1

bÞ;

abpbÞ

b? r2

ð40Þ

tNðp;r;tÞ ¼ tsbtN?1

abðtbapapbÞNtasð1 þ rbarbsp2

þ ðN ? 1Þr2

The paths corresponding to these terms are sketched in

Fig. 5. The transmitted signal immediately behind the wave-

front at s = 0 clearly consists from the waves that have

followed the path without internal reflections in Fig. (5).

Furthermore the contributions from other paths (with inter-

nalreflections) can be neglected for thehigh-frequencycom-

ponents as will be shown now. With Eq. (35), the high-

frequency approximation of the reflection coefficient, Eq.

(38), follows as

bþ ðN ? 1Þr2

aþ ???Þ.

abp2

a

bap2

bþ rasrabp2

ð41Þ

rjj0ðxÞjx2S¼x2

pj0 ? x2

4cos2hs

pj

1

x2þ Oð?4Þð42Þ

with ? from Eq. (28). Here (and in the following) the sum

rule, Eq. (2), has been used. From Eqs. (41) and (42) it fol-

lows that transmitted waves that have experienced reflec-

tions within the N-layer medium give contributions to tN

that are at least quadratic in the reflection coefficients.

Therefore, on the integration path S, they give contribu-

tions of order ?4, so

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

215

Page 6

tNðxÞjx2S¼ tsbtN?1

The Oð?4Þ terms are very small with respect to the wave that

has not experienced internal reflections and may be omitted

when attention is restricted to positions immediately be-

hind the wavefront. The dominant term of Eq. (36a) at

large frequencies is thus

abtN

bataspN

apN

bþ Oð?4Þ.

ð43Þ

tsbtN?1

abtN

bataspN

apN

be?iks

xNK??

x2S’ e

iNa x2

ð

ps?x2

pa

ÞþiNb

2ccoshs

x2

ps?x2

pb

??

1

x.

ð44Þ

At very large frequencies such that jxj ? xM, the spectral

density, Eq. (19), may be expanded to first order as well,

~Es0ðxÞ ¼ ?1

2p

X

m

xm~Es0ðxmÞ

x2

ð45Þ

and the dominant contribution to the initially transmitted

field, Eq. (34) follows as the Sommerfeld precursor

X

where the real function

SsNðs;rsÞ ¼ ?1

2p

m

xm~Es0ðxmÞ

Z

S

dxe?ixs?inðrsÞ=x

x2

;

ð46Þ

nðrsÞ ¼ Nax2

paþ bx2

2ccoshs

pb? Kx2

ps

þx2

ps

2crs.

ð47Þ

For s > 0, the absolute value of the integrand in Eq. (46)

grows exponentially with the radius X of the semicircle part

of S. This does not necessarily imply that the integral di-

verges, since also the oscillation frequency of the integrand

along the semicircle increases. We will next show that the

integral actually converges in the limit X ! 1. The steps

taken in the following are merely a repetition of the work

in [11] in order to perform the integration.

Consider the path?S, which is obtained from path S by

reflection about the point x = 0. On?S, Eq. (46) is arbitrary

small (for arbitrary large X) for s > 0, so this path?S may be

added to S in Eq. (46). The integration over?S is chosen to

run from x = +1 to x = ?1. 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. (46), one

obtains

SsNðs;rsÞ¼1

2p

X

m

xm~Es0ðxmÞ

I

C

dxe

?i

ffiffiffiffiffiffiffiffiffi

nðrsÞs

p

x

ffiffiffiffiffiffi

s

nðrsÞ

p

þ1

x

ffiffiffiffiffiffi

nðrsÞ

s

p

??

x2

.

ð48Þ

Change the integration variable from x to / ¼ ?i

identify the initially transmitted electric field as

X

Here J1is the Bessel function of the first kind and order. Eq.

(49) has exactly the same form as the expression for the

Sommerfeld precursor after transmission through a homo-

geneous medium [11]. In the next section we will discuss the

changes due to the layered structure of the medium.

ffiffis

n

px to

ð49Þ

SsNðs;rsÞ ¼

m

xm~Es0ðxmÞ

ffiffiffiffiffiffiffiffiffiffiffi

s

nðrsÞ

r

J1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffi

nðrsÞs

p

??

.

7. Discussion

First the transmission of a pulse through a homoge-

neous medium and transmission through an N-layer med-

ium will be compared. For transmission over a horizontal

distance NK through a homogeneous medium a and for ini-

tial propagation along the rs-axis in a vacuum surrounding

medium s, the function n in the expression for the initially

transmitted field, Eq. (49), simplifies to

nref? njnb¼na;ns¼1¼

NKx2

2ccoshs.

pa

ð50Þ

Note that the function n of Eq. (47) is symmetric in the

media a and b. So medium b could have played the ro ˆle

of reference medium as well. For the N-layer system in be-

tween vacuum, the function n is

n ¼ Nax2

paþ bx2

2ccoshs

pb

.

ð51Þ

So the plasma frequency squared for a homogeneous med-

ium is simply replaced by the spatial average of the plasma

frequency squared of the N-layer medium.

The influence of the perturbing medium b on the Som-

merfeld precursor will be explicated somewhat further.

The relative slabwidth of slab b is

Fig. 5. First few terms in the path-length ordered form of the transmission coefficient.

216

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

Page 7

b ¼b

K

ð52Þ

and the relative difference of the squared plasma frequen-

cies is

D ¼x2

pb? x2

x2

pa

pa

.

ð53Þ

In terms of these parameters, Eq. (51) reads as

n ¼ ð1 þ dÞnref;

and the Sommerfeld precursor field, Eq. (49), reads as

X

?

ð1 þ dÞnref

Fig. 6 shows the field transmitted field of Eq. (55) for per-

pendicular incidence (hs= 0) and transmission through

several N-layer media with N = 100 K = 600 nm. The refer-

ence medium a is made of silicon which has xpa= 2.4 · 1016

s?1[21, p. 278]. The effect on the precursor field of changing

a homogeneous medium into a photonic structure is cap-

tured in the parameter d. There are three distinguishable

situations:

d ¼ bD

ð54Þ

SsNðsÞ ¼

m

xm~Es0ðxmÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s

r

J1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 þ dÞnrefs

p

??

.

ð55Þ

• d = 0 This corresponds to either zero relative slabwidth

b = 0 or equal plasma frequencies, xpb= xpa= 2.4 ·

1016s?1. This trivial situation represents the homoge-

neous reference medium a and the corresponding ini-

tially transmitted field is given by the solid line.

• d > 0 This is obtained when xpb> xpa together with

nonzero b. The amplitude and period of the initially

transmitted field have decreased with respect to the

transmitted field through the reference medium. This sit-

uation holds for the dotted line ðxpb¼ 3:0 ? 1016s?1>

xpa;b ¼1

• d < 0 When on the other hand xpb< xpaand b 5 0 then

the amplitude and period increase with respect to the

reference medium case, as shown by the dashed line

ðxpb¼ 1:8 ? 1016s?1< xpa;b ¼1

2Þ.

2Þ.

The effects increase with increasing b. The extreme case

d = ? 1 is obtained from putting b = 1 and xpb= 0. The

N-layer medium has reduced to a complete vacuum and

gives for the transmitted field

X

¼

m

lim

d!?1

m

xm~Es0ðxmÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s

nað1 þ dÞ

r

J1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nað1 þ dÞs

p

??

X

~Es0ðxmÞxms.

ð56Þ

This is indeed the approximation for the applied field, Eq.

(16), propagated through vacuum valid for xMs ? 1.

At last, the initial amplitude of the transmitted Sommer-

feld precursor, Eq. (49), is compared to the applied signal

amplitude, Eq. (16). This is done for transmission through

the homogeneous reference medium a over a distance

NK = 6.00 · 10?5m.Forperpendicularincidenceontoasil-

icon dielectric, Eq. (50) gives nref= 5.76 · 1019s?1. The first

maximum of J1is at 2

ns

¼ 1:84 and has the value 0.582.

At this maximum, s = 1.47 · 10?20s. For simplicity take

an applied pulse with only one single optical carrier angular

frequency xM= 3.00 · 1015s?1and amplitude~Es0ðxMÞ,

Es0ðt;rs¼ 0Þ ¼ I½0;T?ðtÞ~Es0ðxMÞsinxMt.

For perpendicular incidence, rs= 0 coincides with the en-

trance line of the medium, x = 0. Hence at the first maxi-

mum, Eq. (49) gives SsN¼ 2:79 ? 10?5~Es0ðxMÞ. Therefore

the initial amplitude of the Sommerfeld precursor for this

specific applied field transmitted through the reference

medium is very small compared to the amplitude of the

corresponding applied pulse.

ffiffiffiffiffip

ð57Þ

8. Conclusion

The wavefront of a pulse that propagates in a dielectric

medium is constructed from the infinite frequency compo-

nents of that pulse, since the electrons of the medium are

inert and no polarization can be induced. When a homoge-

neous dielectric medium is replaced by a rectangular pho-

tonic crystal that consists of layers of the same type of

media, the wavefront therefore still propagates at the vac-

uum speed of light.

Fig. 6. Transmitted field through a homogeneous Si medium (solid line) and through two photonic crystals of the same length, but with different plasma

frequencies.

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

217

Page 8

The Sommerfeld precursor results from very high fre-

quency components of the pulse, which experience the

medium as if it were almost vacuum. This precursor imme-

diately follows the wavefront of the transmitted pulse.

Although the individual frequency components of the

applied pulse experience a frequency-dependent refraction

at each interface within the photonic crystal, after trans-

mission a Sommerfeld precursor emerges of the same shape

as for the homogeneous medium. The only difference is

that in the expression for the Sommerfeld precursor, the

plasma frequency squared for the case of a homogeneous

medium is replaced by the spatial average of the plasma

frequency squared of the layered medium.

Acknowledgements

The authors would like to thank Professor J. Knoester

for his useful suggestions. This research is supported by

NanoNed, a national nanotechnology programme coordi-

nated by the Dutch Ministry of Economic Affairs.

Appendix A

In this appendix the accuracy of the Sommerfeld precur-

sor approximation to the field is estimated in the time

domain. For brevity a plane wave of one single carrier fre-

quency xcis considered which is perpendicularly incident

from vacuum at x < 0 onto a half-infinite homogeneous

medium a at x > 0. This 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

Z

~Lðx;xÞ ¼

q¼0

where s = t ? x/c,~EðxcÞ ¼2

nrðxÞ ? ?x

c

ðr þ 1Þ!

Here gðmÞ ¼ nðxÞjx¼m?1 and n(x) is the refractive index, Eq.

(3), for j = a. Use

Zs

¼

S

ð?iÞnþ1

n!

Eðs;xÞ ¼

S

dx~Lðx;xÞ~SðxÞe?n1ðxÞ=xe?ixs;

X

ðA:1Þ

1

x2q

x2qexp

c

?i

X

1

r¼2

nrðxÞ

xr

!

;

~SðxÞ ¼1

2p

xc~EðxcÞ

x2

;

T

????

RT

0dtEðtÞsinxct and

1drþ1gðmÞ

dmrþ1

m¼x?1¼0

.

ðA:2Þ

ð?iÞnþ1

n!

0

ds0ðs ? s0ÞnSðs0;xÞ

1

xnþ1~SðxÞe?ixs?in1ðxÞ

Rs

dxe1=xnþ1~SðxÞe?ixs?in1ðxÞ

Z

dx

?

x;

ðA:3Þ

Sðs;xÞ þ

e

0ds0ðs?s0Þn? 1

?

Sðs0;xÞ

¼

Z

S

x;

ðA:4Þ

to rewrite Eq. (A.1) as E(s,x) = S(s,x) + (L*S)(s,x) where

convolution is with respect to the variable s and

L = ?1+ ?2+ ?1*?2with

?1? Sðs;xÞ ¼

X

1

q¼1

x2q

c

ð?iÞ2q

ð2q ? 1Þ!

Zs

0

ds0ðs ? s0Þ2q?1Sðs0;xÞ;

ðA:5Þ

?2ðxÞ ? Sðs;xÞ ¼

exp

?i

X

1

r¼2

nrðxÞð?iÞr

ðr ? 1Þ!

?

Zs

0

ds0ðs ? s0Þr?1

!

? 1

!

Sðs0;xÞ.

ðA:6Þ

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 ? 2 R,

kRk2¼ kL ? Sk2< ?kSk2;

where kRk2is the L2-norm of the function R on the interval

(0,s),

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s

0

ðA:7Þ

kRk2¼

1

Zs

ds0jRðs0;xÞj2

s

.

ðA:8Þ

Since L is linear, its norm is (see [22]) kLk = supkfk=1kLfk.

Therefore kLfk 6 kLk and we may require kLk2< ?. This

requirement is relaxed a little by demanding

k?1k2< ?=2and

k?2k2< ?=2

ðA:9Þ

and neglecting the ?1*?2-term. A typical term in L is of the

form

Zs

2

Zs

Zs

2

Zs

The supremum in the right-hand-side of Eq. (A.10) is

found by demanding the functional

Zs

1

s

0

ð?iÞnþ1

n!

0

ds0ðs0? sÞn

?

?nþ1

?????

?????

2

¼?i

0

ds1

?

?

?i

?????

Zs1

2

0

ds2

??

??? ?i

Zsn

0

dsnþ1

??

????

????

2

2

6

?i

0

ds0

?????

kfk¼1

¼ sup

1

s

0

ds0

Zs0

0

ds00fðs00Þ

!2nþ2

.

ðA:10Þ

I½f?ðsÞ ¼1

s

0

ds0

Zs0

Zs

0

ds00fðs00Þ

!2nþ2

þ kðsÞ

ds0fðs0Þ2? 1

??

ðA:11Þ

to be stationary under variations in f. Here k is a Lagrange

multiplier. This gives f = ±1 and

Zs

2

Therefore

ð?iÞnþ1

n!

0

ds0ðs0? sÞn

?????

?????

6

snþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2n þ 3

p

.

ðA:12Þ

218

R. Uitham, B.J. Hoenders / Optics Communications 262 (2006) 211–219

Page 9

k?1k26

X

1

q¼1

x2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

cs2q

4q þ 1

1

p

and

k?2k26 exp

r¼2

jnrðxÞjsr

ffiffiffiffiffiffiffiffiffiffiffiffiffi

2r þ 1

p

!

? 1.

ðA:13Þ

This gives

X

so if

1

q¼1

x2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cs2q

4q þ 1

p

< ?=2;

X

1

r¼2

jnrðxÞjsr

ffiffiffiffiffiffiffiffiffiffiffiffiffi

2r þ 1

p

< ?=2.

ðA:14Þ

xcs < ?=2and

jn2js2¼cx2

pax

2c

s2< ?=2;

ðA:15Þ

then requirement Eq. (A.9) is fulfilled and the Sommerfeld

precursor gives an accurate description of the electric field.

For the choice xc= 3.00 · 1015s?1the inequality on the

left-hand-side of Eq. (A.15) gives s < 0.17? fs. With

c = 4 · 1013s?1, xpa= 2.4 · 1016s?1and x = 6 · 10?5m

the other inequality gives s < 1:4 ? 10?17

0.01 and with oscillation times of ?10?19s (see Fig. 6) the

approximation is accurate over ?102oscillations.

ffiffi?

p

s. So for ? =

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219

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