omnidirectional reflection from a slab of uniaxially anisotropic negative refractive index materials
TRANSCRIPT
Optics Communications 240 (2004) 19–27
www.elsevier.com/locate/optcom
Omnidirectional reflection from a slab of uniaxiallyanisotropic negative refractive index materials
Zheng Liu *, Jianjun Xu, Zhifang Lin
Department of Physics, Research Center of Theoretical Physics, Fudan University, Handan Road 220, Shanghai 200433, PR China
Received 30 December 2003; received in revised form 19 May 2004; accepted 8 June 2004
Abstract
In this paper, we explore the reflection properties of multilayer containing a slab of uniaxially anisotropic materials
with negative refractive index. The conditions are discussed under which the omnidirectional reflection can occurs in a
certain frequency region. In addition, we also investigate the photonic band gaps (PBGs) of the one-dimensional lay-
ered stack that consists of uniaxially anisotropic slabs. It is found that the PBGs is invariant upon a change of scale
length and is insensitive to disorder of the layer thickness.
� 2004 Elsevier B.V. All rights reserved.
PACS: 78.20.Ci; 78.66; 41.20.Jb; 42.25.Bs
Keywords: Omnidirectional reflection; Negative refractive index
1. Introduction
Recently much experimental and theoretical ef-
forts has been devoted to the study of a new type
of metamaterials that have simultaneously nega-
tive electric permittivity � and magnetic permeabil-
ity l and thus termed as negative refractive index
0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv
doi:10.1016/j.optcom.2004.06.019
* Corresponding author. Tel.: +86215664159401; fax:
+862165104949.
E-mail address: [email protected] (Z. Liu).
materials (NIMs), or ‘‘left handed materials’’[1–11]. This type of materials are usually construct-
ed by a periodic array of split ring resonator, which
are placed in the second medium of interacting
wires. When two effective medium characterized
by effective leff(x) and �eff(x), respectively, are
combined, there would be a frequency region with-
in which the effective permittivity and permeability
of the combined medium are simultaneously nega-tive [4,5]. The term ‘‘left handed’’ comes from the
fact that the electric field E, magnetic field H and
wave vector k form a left-handed triplet of vectors.
ed.
20 Z. Liu et al. / Optics Communications 240 (2004) 19–27
The materials with negative refractive index usu-
ally have many unique properties such as the rever-
sal of the Doppler shift for radiation, the reversal
of Cherenkov radiation and the anomalous refrac-
tion. So recently great interests have been focusedon exploring the unusual electromagnetic effects
of NIM that may lead to potential application.
Among others, ideas have been proposed which
use NIMs to achieve subwavelength focusing
[7,8], to open an unconventional photonic band
gap [9], to enhance photon tunneling [10,11] and
to approach light localization [12]. Here, we
explore its unique property of negative values inpermeability and permittivity to produce omnidi-
rectional reflection.
Omnidirectional reflectors (ODRs) is the kind
of optic dielectric structure, which can reflect the
electromagnetic wave incident at any angles with
any polarization and has drawn much attention re-
cently [13–19]. Generally there have been two
types of reflectors that may produce omnidirec-tional reflection. The first one is the age-old metal-
lic reflectors. A slab of metal can reflect the light
over a wide range of frequencies for any incident
angle and polarization. But it suffers from loss at
infrared frequencies. The second one is ordinary
dielectric multilayers, which was first proposed
by Joanopoulos [13–15] based on the one-dimen-
sional (1D) photonic band gaps (PBGs) and re-ceived much improvement, including structure
involving anisotropic stacks, over last few years
[15–19]. Here we propose the third type of reflector
that producing omnidirectional reflection, a slab of
uniaxially anisotropic materials with negative re-
fractive index.
Besides, the PBGs material have also absorbed
tremendous interest in the last decade due totheir practical importance. Many work has been
devoted to it. So far two types of PBGs� havebeen reported. One is the conventional PBGs in-
duced by Bragg scattering [20]. The other is the
zero-�n gap as the result of zero (volume) aver-
aged refractive index [9]. In the latter part of this
paper the PBGs produced by the 1D stack con-
taining uniaxially anisotropic layers are also dis-cussed, which are based on the different
mechanism from Bragg scattering and zero-aver-
aged index refraction.
2. Structure and its properties
Our composite system for omnidirectional re-
flection is composed of one or two slabs placed
in the isotropic background and one of the slabsis made of uniaxially anisotropic negative index
material. Firstly, we consider the case that there
is only a uniaxially anisotropic slab in the back-
ground. we denote the permittivity and permeabil-
ity of the background as �0, l0 and take the slab�soptical axis as the z axis, the permittivity and per-
meability tensors of the slab have the following
forms:
� ¼�t 0 0
0 �t 0
0 0 �z
0B@
1CA; l ¼
lt 0 0
0 lt 0
0 0 lz
0B@
1CA;
where �z, (lz) and �t, (lt) are the permittivity (per-
meability) constants in the directions parallel and
perpendicular to the optical axis, respectively.
Since the media are isotropic in the x–y plane,
we can assume that the wave vectors are in thex–z plane without loss of any generality, Based
on the Maxwell equations the dispersion relations
of the uniaxially anisotropic medium can be
worked out as follows [21]:
k2xx2�tlz
þ k2zx2�tlt
¼ 1 for E polarization; ð1Þ
k2xx2lt�z
þ k2zx2lt�t
¼ 1 for H polarization: ð2Þ
For the plane wave incident on the interface the
tangential component kx of its wave vector is con-
tinuous across the layers according to the bound-
ary conditions. For a fixed kx, the solution to kzfrom Eqs. (1) and (2) can be either real or imagi-
nary. For the imaginary kz, the magnitude of the
field will decay exponentially and the wave is re-flected totally by the interface. For some given �t,�z, lt and lz, there would exist a value of kx for
which k2z ¼ 0 and we denoted that kx as kc. If
k2z < 0 happens when kx> jkcj, namely, the wave
in the slab is evanescent when kx> jkcj, it is calledcutoff. On the contrary when kx < jkcj; k2z < 0 oc-
curs it is called anti-cutoff [8].
Z. Liu et al. / Optics Communications 240 (2004) 19–27 21
From Eqs. (1) and (2), it follows that if inequal-
ities
�tlz > 0; �zlt > 0; �tlt < 0; ð3Þor
�tlz < 0; �zlt < 0; �tlt > 0 ð4Þare satisfied, the dispersion curve of the slab are
hyperbolas for both E and H polarization. The
former have the same semiminor axis and the lat-
ter have the same semimajor axis and both of themare parallel to the optical axis.
Firstly, let�s consider the case that there is one
uniaxially anisotropic slab in the background,
which corresponds to the anti-cutoff. As shown
Fig. 1. The permittivity and permeability of the uniaxially anisotropi
and H polarization are two hyperbolas with the same semimajor.
background. (c,d) Correspond to the cases that a second isotropic sla
hyperbola�s semimajor is added.
in Fig. 1(a) only if the radius xffiffiffiffiffiffiffiffiffi�0l0
pis less than
the length of their semimajor axis: xffiffiffiffiffiffiffiffi�tlz
p(for E
polarization), xffiffiffiffiffiffiffiffi�zlt
p(For H polarization) or
xffiffiffiffiffiffiffi�tlt
p(For both E and H polarization) i.e.,
n20 � �0l0 < �tlz; ð5Þ
�0l0 < �zlt; ð6Þ
where n0 is the refractive index of the background
or
n20 < �tlt � n2t ð7Þ
the kx will be less than kc and the wave in the slab
will become evanescent for all the incident angle
c slab satisfy �tlz>0, �zlt>0, �tlt<0 the dispersion curve for E
(a,b) Correspond to the cases there are only one slab in the
b �1 Æl1 with the radius of the dispersion circle less than the the
22 Z. Liu et al. / Optics Communications 240 (2004) 19–27
and thus the omnidirectional reflection has been
obtained.
On the contrary, if the inequalities (5)–(7) are
not satisfied kx will be larger than the kc and the
wave will be transmitted though the slab for the in-cident angles that are larger than the critical angleffiffiffiffiffiffiffiffiffi
�tlz
�0l0
r;
ffiffiffiffiffiffiffiffiffi�zlt
�0l0
ror
ffiffiffiffiffiffiffiffiffi�tlt
�0l0
r
as shown in Fig. 1(b). To offset the angle hole an-
other slab is necessary the dispersion curve of
which is close like a circle, a ellipse. This corre-
sponds to the second case that there are double
slabs in the background one of which is uniaxially
anisotropic and the other is assistant. Only if the
dimension of the close curve is less than the length
of the hyperbola�s major, as shown in Fig. 1(c) or(d), the total reflection would always occur
whether the incident angle is larger than the criti-
cal angle or not. It is because that the wave inci-
dent from the background dielectric always
experience the cutoff or the ani-cutoff in spite of
the incident angle. For example, in the subcase
(c) the two slabs provide the series of cutoff, ant-
cutoff. Here the assistant layer is isotropic withthe permittivity �1 and permeability l1. The omni-
directional reflection conditions under the above
inequalities (3) and (4) are
�tlz>Minð�0l0;�1l1ÞðEÞ; �zlt>Minð�0l0;�1l1ÞðHÞð8Þ
for the inequality (3) with the interface perpendic-
ular to the optical axis or
n2t ¼ �tlt > Minð�0l0; �1l1Þfor both E and H polarization; ð9Þ
for the inequality (4) with the interface parallel tothe optical axis.
Generally, the NIMs are highly dispersive and
the �, l can vary from very negative values to very
positive ones with the frequency and the inequali-
ties for the uniaxially anisotropic media are not al-
ways satisfied for all the frequency spectrum. So
there usually exist some discrete frequency win-
dows in the spectrum where the inequalities aresatisfied. Let�s consider a uniaxially anisotropic
slab with most frequently adopted effective param-
eters [22] forms
�tðf Þ ¼ 1þ 20
3:782 � f 2þ 100
122 � f 2; ð10Þ
lzðf Þ ¼ 1þ 7
3:82 � f 2ð11Þ
and �z=lt=1, where f denotes the frequency mea-
sured in GHz. Such resonant forms as Eq. (10) and
(11) are typical in NIM [9,12,22] and can be
achieved experimentally by using a network of thin
wires and a periodic arrangement of split ring res-
onators, respectively. Both of them have been
adopted for numerical calculation of 1D photonic
band of layer heterostructures combining PIMsand NIMs [9]. The refractive index of the back-
ground is 1.33 e.g. water or ice and the assistant
layer is air with �1=l1=1. It can be easily found
that in the frequency range 3.8–4.63 GHz the per-
mittivities �t, lz are simultaneously negative and
therefore �tlz>0, �zlt>0, �tlt<0, and further-
more, in the range of [3.8, 4.39] the product �tlzclose to unity from the very positive large valuewith �tlt<0, which satisfy (8). In the range of
[4.39, 4.63], 0< �tlz<1 with �tlt<0, which does
not satisfy (8). On the other hand, in the range
of [4.63, 5.01] �t<0, lz>0 with �tlt<0 and this
case corresponds to the always cutoff. So from
these analyses we can draw a conclusion that there
exist two frequency windows at least in the spec-
trum on the optical structure where the omnidirec-tional reflection can be obtained. Here we
numerically calculated the reflectance of 1D lay-
ered stack consisting of 10 periods with transfer
matrix method where there are an air layer and a
NIM layer, respectively, in a period for different
incident angle. As shown in Fig. 2. it can be found
that no matter what the incident angles are the re-
flectance in the range [3.8,4.39][ [4.63, 5.01] is al-ways close to unity, i.e., the gap in the interval is
invariant to the incident angle whether its width
or its position in the spectrum, while other Bragg
gaps in which the reflectance also approach to uni-
ty are changed and destroyed when the incident
angle varies.
In fact, when the frequency in the gap is small
the �tlz is very large, e.g., 199.86 at frequency3.89 and extremely larger than �0l0 or �1l1 and
the above omnidirectional conditions are easily
(d)(c)
(b)(a)
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0θ
0=890
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0θ
0=660
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0 θ0=450
Frequency (GHz) Frequency (GHz)
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0θ
0=00
Ref
lect
ance
Ref
lect
ance
Fig. 2. The reflectance of 1D NIM layered stack consisted of 10 periods for different incident angle. There are a air layer and a NIM
layer, respectively, in a period a. The ratio is 0.8:0.2.
Z. Liu et al. / Optics Communications 240 (2004) 19–27 23
satisfied. This corresponds to Fig. 1(a). With the
increase of the frequency in the gap range, the
length of the hyperbola�s semimajor rapidly
shrinks and for some larger incident angles the
electromagnetic wave will be transmitted if there�sno assistant layer like Fig. 1(b). Once the assis-
tant layer is sandwiched with its refractive index
less than the background�s it will reflect the waveswith larger incident angle like Fig. 1(c) or (d). So
the interval of invariant frequency is between the
value at which the length of the hyperbola�ssemimajor is close to infinity and the value at
which the semimajor�s length is equal to the
radius of the dispersion circle of the assistant
layer.
Besides the stability to the incident angle the
NIMs gaps produced by the layered stack of
NIM are also stable and not sensitive to the total
thickness and the disorder of the ratio of layers�thickness in a period just because it�s based on
the cutoff principle. The conventional ODRs are
usually manufactured underlying physics of Bragg
scattering and the omnidirectional gap is producedby the periodic system [19]. The gaps of this type
of ODRs is sensitive to the disorder of the ratio
of two layer�s thickness as well as the incident an-
gle. As shown in Figs. 3(a) and (b) the transmit-
tance of a 1D isotropic layered stack has also
been calculated for different lattice constant a
and different degree of the disorder in thickness.
(b)
(a)
0.2
0.4
0.6
0.8
1.0
Tra
nsm
ittan
ceT
rans
mitt
ance
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Frequency (GHz)
Fig. 3. (a) Solid line: transmittance through 10 U cells with lattice constant a, dashed line and dotted line correspond to the
transmittances of 10% and 20% disorder. (b) Transmittance when the lattice constant is scaled by 2/3.
24 Z. Liu et al. / Optics Communications 240 (2004) 19–27
It consists of ten periods with the refractive indices
of the two isotropic layers n1=1.6, n2=4.6. The
thickness parameters of the two layers are
d1=0.8a, d2=0.2a. In Fig. 3(a) the solid line corre-
sponds to a perfectly ordered stack with lattice
constant a and the dotted line and dashed line cor-
responds to the thickness disorder (random uni-
form deviate) of 10% and 20% respectively, eachensemble averaged over 50 realizations. Fig. 3(b)
corresponds to the transmittance through the same
materials, but the lattice constant a scaled by a fac-
tor of 2/3. It�s easily found that the Bragg gap is
destroyed by strong disorder in (a) and shifts be-
cause of the lattice constant�s being scaled in (b).
However, the range of ODR for structure consist-
ing of NIM is invariant and not sensitive to thesefactors whether the strong disorder or the total
thickness� being rescaled as shown in Fig. 4. In
Fig. 4(a) the dotted line corresponds to transmit-
tance of the layered stack of NIM with the same
total thickness 10a. The solid line corresponds to
thickness disorder (random uniform deviate) of
30% and Fig. 4(b) gives the transmittance which
corresponds to the lattice constant a scaled by
the factor of 2/3 with the same ratio (0.8:0.2) as
the one of the dotted line in Fig. 4(a). From Fig.
4 it can be found that the gaps [3.8, 4.39][ [4.63,
5.01] exist all the time in spite of these factors,which is due to the (anti-)cutoff principle in stead
of Bragg reflection.
There is another subcase for the anisotropic
slab that �tlz<0, �zlt<0, �tlt<0 which also corre-
sponds to the imaginary kz for any kx. it is called
always cutoff. For this case there is no special re-
quirement for the refractive index of the back-
ground to obtain the omnidirectional reflection.However, it requires all three component of � or
l be negative, such a system of always cutoff is dif-
ficult to make because currently negative �; l can
only be achieved for particular polarization.
(b)
(a)
20.
40.
60.
80.
01.
0 2 4 6 8 1000.
20.
40.
60.
80.
01.
Frequency (GHz)
Tra
nsm
ittan
ceT
rans
mitt
ance
Fig. 4. (a) Dotted line, transmittance of 1D NIM layered stack with 10 periods for normal incident; solid line, the transmittance of the
stack with 30 averaged over 50 realizations. (b) Transmittance when the lattice constant is scaled by 2/3.
Z. Liu et al. / Optics Communications 240 (2004) 19–27 25
Furthermore, because of the metals contained
in the left-handed media the effect on the gaps ex-
erted by the absorbing nature of the media inevita-
bly deserve to be estimated. To explore the effect
we add some imaginary parts to Eqs. (10) and
(11) to phenomenologically characterize the ab-
sorbing nature. The modified forms are as follows:
�tðf Þ ¼ 1þ 20
3:782 � f 2 � ic1fþ 100
122 � f 2 � ic2f;
ð12Þ
lzðf Þ ¼ 1þ 7
3:82 � f 2 � iCf: ð13Þ
For Eq. (12) the plasma frequency fp1, fp2 are
4.47, 10 GHz and for Eq. (13) the resonance fre-
quency f0 is 3.8 GHz. We have calculated the re-
flectance of the stack same as the case in Fig. 2.
except that the dispersion relations are replaced
by Eqs. (12) and (13) for the fixed incident angle
45� with different damping parameters ci and C(i=1, 2). The results calculated shown in Fig. 5.
imply that the position of the gaps are affected less
by the absorbtion of the media than the amplitude
of the reflectance. The system keeps effective for
the small parameters but not for the larger ones.
So the NIMs with low loss are expected for the
purpose of the practical application of the typeof ODRs.
3. Summary
We have explored the reflection properties of
the multilayer including a slab of uniaxially aniso-
tropic negative index materials. To the fixed fre-quency we have gotten some inequalities on the
dielectric parameters of the layers for the omnidi-
rectional reflection according to the cutoff, anti-
cutoff and always cutoff concept which is drawn
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Ref
lect
ance
0.00 0.01 0.02 0.03
Frequency (GHz)
Fig. 5. The reflectance of 1D NIM layered stack same as in Fig. 2 for different damping parameters with the fixed incident angle 45�.The damping parameters ci, C are assumed as fpi, f0 multiplied by 0, 0.01, 0.02, 0.03, respectively, where i=1, 2. For Eq. (12) the plasma
frequency fp1, fp2 are 4.47, 10 GHz and for Eq. (13) the resonance frequency f0 is 3.8 GHz.
26 Z. Liu et al. / Optics Communications 240 (2004) 19–27
out from the phase matching conditions [23] on
each interface. Considering the highly dispersive
property of the negative index materials we de-duced the existence of the gaps in which the elec-
tromagnetic wave will be totally reflected for any
incident angle. Furthermore, this type of gap
based on the underlying mechanism different from
Bragg reflection has been found to possess simulta-
neously a serials of stability, i.e., the width and the
position of the gap are not sensitive to the incident
angle, the disorder of the total thickness and theratio of each layers� thickness. Because of the ex-
tremely large value of the parameters within the
gaps the foregoing inequalities can easily be satis-
fied and the kinds of the background dielectric
suitable to the type of the omnidirectional reflector
will be extensive. Similarly because of the large
value of the parameters the skin depth will be very
small and therefore the thickness needed for thistype of ODRs is smaller than the one of the ODRs
underlying the Bragg reflection. The absorption of
the media brings some limitations to the applica-
tion of the ODRs to some extent. We may expect
this disadvantage can be overcomed with the pro-
gress of the technology of materials.
Acknowledgement
This work was supported by CNNSF andCNKBRSF.
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