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
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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
1 for
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< ?=2 and
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 < ?=2 and
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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