Visualization of Negative Refraction in Chiral Nihility
Media
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Xiangxiang Cheng et al. “Visualization of Negative Refraction in
Chiral Nihility Media.” Antennas and Propagation Magazine,
IEEE 51.4 (2009): 79-87. © 2009, IEEE
As Published
http://dx.doi.org/10.1109/MAP.2009.5338687
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Institute of Electrical and Electronics Engineers
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Thu May 26 19:16:15 EDT 2016
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Detailed Terms
Visualization of Negative Refraction in
Chiral Nlhlllty Media
Xiangxiang Cheng 1, Hongsheng Chen 1,2, Bae-Ian Wu2, and Jin Au Konr/'*
Electromagnetics Academy at Zhejiang University
Zhejiang University, Hangzhou 310058, China
1The
2Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
E-mail: [email protected]
*Deceased
Abstract
Chiral nihility media are known to produce a backward wave as one of their two polarizations, yielding negative refraction at
the interface of the chiral nihility medium and free space. In this paper, the analysis of chiral nihility effects is illustrated by
showing the propagation of a Gaussian beam, both reflected and refracted from an air-chiral interface, and through layered
chiral nihility media that are matched to free space. The critical angle for total reflection, and the index-matched total
transmission in a matched chiral half-space, are demonstrated. Also demonstrated are the wave splitting, wave widening, and
a wave of "standing phase" in matched chiral nihility slabs.
Keywords: Chiral media; nihility; polarization; Gaussian beams; matched media; index matched; wave splitting;
electromagnetic refraction; negative refraction
1. Introduction
R
ecently, we have seen evidence of the fast development of the
study and design of materials with negative electromagnetic
parameters supporting backward waves, which are represented by
popular metamaterials. As a matter of fact, backward waves can
exist in more general linear media, namely in bianisotropic media,
such as chiral media [1, 2], and in gyrotropic media [3, 4], which
belong to different subsets of magneto-electric materials. Chiral
media, which were discovered in the beginning of the 19th century,
have been of much scientific interest. They have many practical
applications for scientists and engineers in many different fields
(e.g., physics, chemistry, and biology). The well-known natural
phenomenon - optical activity - related to chiral media can rotate
the polarization of an incident linearly polarized wave when propagating through a chiral medium. The polarization state of the
transmitted linearly polarized wave depends significantly on the
constitutive parameters' values, as well as on the thickness of the
chiral medium and the incident angle [5-7].
A special case of chiral media with both permittivity and permeability equal to zero is referred as a "chiral nihility" medium [8].
This is not physically realizable in electromagnetic materials [9,
10]. Hence, in [10], the author proposed a modified condition of
chiral nihility, in which the permittivity and permeability tend
toward but are not equal to zero, to which we will adhere in our
paper.
IEEE Antennasand Propagation MagaZine, Vol. 51, No.4, August 2009
Before chiral nihility, the concept of nihility, the electromagnetic nilpotent, was introduced, in which wave propagation cannot
occur, and thus the directionality of the phase velocity relative to
the wave vector is a non-issue [11, 12]. Clearly, nihility is
unachievable, but it may be approximately simulated at a specified
frequency [13, 14]. The properties of matched zero-index passive
metamaterials presented in [13] demonstrated that in the steady
state, the fields inside the media are static in character, but the
underlying dynamics are those of propagating transverse waves.
The zero index means that the index of refraction is not matched to
free space although the impedance is matched. However, in chiral
nihility media, both would be matched, and the directionality of the
phase velocity relative to the wave vector would not be a non-issue
any more, because waves propagate in chiral nihility media.
In this paper, we investigate the propagation of a Gaussian
beam in a two-layer (in Section 2) and a three-layer (in Section 3)
setup, the second layers of which are both chiral nihility media that
are matched to free space. Different polarizations of the incident
waves are considered, respectively. The waves inside and outside
the chiral media are solved for analytically from Maxwell's equations by matching the boundary conditions at the interfaces. It is
shown that the fields in all regions of both cases can be unambiguously calculated in closed form. Some unique characteristics of the
chirality effects on energy reflection and transmission - such as
total reflection, transparency, and wave splitting, which are
strongly dependent on the chirality value - are observed.
ISSN 1045-9243/2009/$25 ©2009 IEEE
79
2. Chiral Half Space
Upon the LHCP wave incidence, two reflected waves (LHCP,
RHCP), and two transmitted waves (LHCP, RHCP), are expected.
The total reflected and transmitted electric and magnetic fields are
expressed as follows:
The constitutive relations of isotropic chiral media are [5]
t , = f dkzl/f (k z)[ R- (el - e2i)e -ikixx+ikzz
<:IJ
(1)
-00
(4)
where ;
=;oJEoJJo .
A monochromatic time-harmonic variation
exp ( -iOJt) is assumed throughout this paper, but omitted. With the
application of a systematic approach named the kDE system,
described in [5], two characteristic waves along with their polarization states inside the isotropic chiral media are analytically
solved for. These include one right-hand circularly polarized
(RHCP) wave, with wavenumber
fir
=
1
-00
dkzl/f(kz)[R- e\i+e2 e-ikixX+ikzz
770
+R+ -eli +e2 e-ikixX+ikzz] ,
770
t; =
1
dkzl/f (k z)
(5)
[r-(e\ - e2"i) eik;;x+ikzz
-00
and the other left-hand circularly polarized (LHCP) wave, with
wavenumber
(6)
fit
They propagate at different phase velocities of OJ/ k + and
Both waves possess the same impedance, which is
+
-
77 = 77 =
OJ/ k -
.
r; = 77 •
V-;
=
1
dkZl/f(kz)[r- e\i+e2" eik;;x+ikzz
-00
77
+r+ -e\i i eik;X+ikzz
77+e
J.
(7)
in which e2' e2" , and ei are unit vectors of the kDE systems when
The + sign refers to right-hand circularly polarized (RHCP)
waves, and the - sign refers to left-hand circularly polarized
(LHCP) waves.
Consider a half-space of an isotropic chiral medium, as
shown in Figure 1, with the following incident LHCP wave :
the k vectors are pointing in different directions (which means
different e3 directions) :
z
(2)
where
(3)
is the Gaussian spectrum that carries the information about the
shape of the footprint centered at x = 0, z = 0 [15]. (At x = 0 , the
amplitude at
=g is 1/e times that of the center .) e\ , e2 ' and e3
Izi
x
are the unit vectors of the kDE system, with e3 always lying in the
direction of
k = e3k . Here, the incident beam is cenk; =xkix + zkiz =xkocos 0i + zkosin 0i' where 0i is the
k
such that
tered about
incident angle of the central plane wave, so that we have
E,
u, ~o
and
-i
•. () •
e2
=xsm
i - z COS()i
80
:
Figure 1. LHCP wave transmission and reflection at an isotropic chiral half-space.
IEEEAntennasand Propagation Magazine, Vol. 51, No.4, August 2009
-10
6
-10
3.5
5
-5
,..<
"'N
a
3
,..<
"'N
2.5
a
2
1.5
2
5
3
-5
4
5
1
1
1_Qo
0
xl>..
0.5
1_~0
10
a
xI'A
10
(b)
(a)
-10
-5
1.5
,..<
"'N
,..<
"'N
1.5
a
1
1
5
a
xl>..
10
(e)
1_~0
0.5
a
xI'A
10
(d)
FIgure 2. The time-averaged power density for a left-hand circularly polarized Gaussian beam incident upon an isotropic
chiral nihility half-space. (a) ~o =0.25. (b) ~o =0.5 . (c) ~o =1. (d) ~o =5. The units for the time-averaged power density
shown are mW 1m 2 •
IEEE Antennasand Propagation Magazine, Vol. 51, No.4, August 2009
81
~ =1x 10-5 co' P
=1x 10-5 PO '
with different chirality values. The
beam propagated at an incident angle of OJ = 25° , and the electric
field of the central incident wave was assumed to have a magnitude
of 1 V1m. The unit for the time-averaged power density shown was
ei =xsinOt -zcosot ·
The k~ are the x components ofwavenumbers for different characteristic waves inside the chiral medium :
mW/m 2 . As predicted, Figure 2c shows the case of total transmission when the refractive index of the chiral nihility 's LHCP wave
is very close to -1 . Under other chirality values, there always were
LHCP reflections. Total reflection happened in Figure 2a, when the
chirality, ';0' was small enough to make the critical angle
where the 0l± are the refracted angles for both circularly polarized
waves, which can be given by
(Oe
n±
UI
. kosin OJ
=arcsin
--+- .
(8)
k-
The reflection and transmission coefficients for both circularly
polarized waves, R± and t", can be found by matching the
boundary conditions for the tangential (.y and z) electric and magnetic fields at the x = 0 interface. When the LHCP wave impedance matching condition, 17follows :
=170 '
is satisfied, the results are as
(9)
R- = cos OJ - cos 01cos OJ + cos 0tT"
=
(10)
2 cos OJ
cos OJ + cos 01-
(11)
These equations indicate that RHCP wave will not be excited,
since the incident wave is also a LHCP wave. However, there will
be a reflected LHCP field, unless it happens that either
e,- =OJ or
leads to R" =0 and T" =1 [2]. In chiral nihility
media, while the impedance-matching condition is fulfilled, the
total transmission happens when ';0 =±1, which likewise makes
the index match to free space.
01-
=-OJ. This
The time-averaged power density can be expressed as
(12)
where
= 14.48°)
of the LHCP wave smaller than the angle of inci-
dence. In the case of Figure 2b, the critical angle (Oe =30°) was
larger than the wave's incident angle, so the beam could be transmitted into the chiral half space, but not totally transmitted. Along
with this, Figure 2d also shows that neither total reflection nor total
transmission occurred as ';0 grew larger than one. Although the
reflection amplitudes were very small compared to the transmission amplitudes in Figure 2b and Figure 2d, instead of directly
seeking the reflected waves, we could easily still find them through
their trails in the interference stripes on the air side, near the halfspace boundary.
3. Chiral Slab
In this section, the propagation of a Gaussian beam through
an infinite chiral slab of thickness d in free space is considered.
The configuration is shown in Figure 3. For a pure circularly
polarized incident wave (LHCP or RHCP) and under the impedance-matching condition , the other circularly polarized wave
(RHCP or LHCP) will not present itself while all three regions are
occupied by only one characteristic wave with the same polarization as the incident wave. This is similar to what we have discussed
in the previous section.
However, in this section we investigate the case when the
Gaussian beam is the same as in the previous section, but with a
linear polarization. Upon TE incidence (or perpendicular polarization [10], or y, in our case), the reflected and transmitted fields
contain both TE and TM components. This is due to boundary
conditions, and the fact that waves should be circularly polarized
inside the chiral slab. Using the same method and expressions in
the previous section, the total electric and magnetic fields upon
z
E,
u., ~o
RHCP
x
LHCP
Equation (12) is for arbitrary cases, once all the complex fields in
each region are known .
Figure 2 illustrates the time-averaged power density as a
function of x and z for the impedance-matched cases where
82
Figure 3. The configuration for a chiral slab of thickness d
placed in free space.
IEEE Antennasand Propagation Magazine, Vol. 51, No.4, August 2009
-20 r---
y
-r-- ,r --
-
RCP E Distribution (VIm)
RCP E Distribution (VIm)
RCP E Distribution (VIm)
y
-,
-10
-20 r----r-~I"""'"'~~rw;I
-20 _ -
-10
- 10
~
o
y
-.--
. -.---
""'1
~
-0.2
-OA
-0.5
0 xf)... 10
0 xf)... 10
LCP E Distribution (VIm)
LCP E Distribution (VIm)
y
0.5
~
0
o xl)..
20
10
~
02
o
o
-0.2
-0.2
-OA
-0.4
-20_ 0.5
-10
~
0
20
10
Total E Distribution (VIm)
y
-r"'...",..F""'..............~
-20..-0.5
-10
o
~
y
-...-
..--
-
--n
0.5
- 10
o
~
-0.5
-0.5
-1
20
10
(f)
Total E Distribution (VIm)
y
xf)...
02
(e)
__- . _ . - - - _
0
OA
o w»;
Total E Distribution (VIm)
1_~0
OB
OA
-0.5
(d)
-20 _ -
y
~O
-10
-10
LCP E Distribution (VIm)
y
-20
-0.5
10
(e)
(b)
(a)
-20
o xl)..
20
-0.5
-1
1_~0
0
v I)..
(g)
10
o
20
(h)
-20
-1
'i f).. 10
(i)
-20
1.2
-15
-15
-10
-10
~ -5
~ -5
o
0.4
o
5
0.2
5
o
xf)...
10
o
20
(j)
time-averaged power density
.u =Po , ; 0 =0.5 .
10
20
(I)
Figure 4. (a)-(c) The RHCP E y component; (d)-(t) the LHCP
space. (a, d, g, j): &=l xlO -
xf)...
s,
component; (g)-(i) the total Ey component; and (j)-(l) the
1(5')1 for an obliquel y incident Gaussian beam upon three different isotropic chiral slabs in free
5&0,
.u =l xlO - 5 Po, ;0 =0.7 . (h , e, h, k): &=l xlO - 5&0 , .u=l xlO- 5 Po, ; 0 = 0.5 . (c, f, i, I): &= &0'
incidence can be computed. The electric fields are expressed as
follows:
(17)
In free space,
(18)
(13)
r. = f dkzlJf(kz)[ R- (e\ -e2i )e-ikuX+ikzZ
ei2k; d
00
+
B- =
cos ().(cos
0..I - cos ()± )
I
DE
-00
+
T- =
Et =
f dkzlJf (kz )[ r: (e\ - e~i ) eiktxx+ikzz
2e
i(-k. +k±)d
IX
x
(-1 +e i2ki d
DE =
=kix
i)
2{1+ i 2ki d
i)
)cos
2
()±.
(21)
Examples for Figure 3 are given in Figure 4, where the incident angle of the Gaussian beam was 25° and the thickness of
chiral slab was d = 6l, where l is the wavelength of the incident
wave in free space. Figure 4 shows the RHCP E y component (Fig-
+A+(e\ + e~+ eik; x+ikzz + B- (e\ - et- e-ik;x+ikzz (16)
ures 4a to 4c); the LHCP E y component (Figures 4d to 4f); the
+B+(e\ +e~i)e-ik;X+ikzz
total Ey component (Figures 4g to 4i); and the time-averaged
J.
ei- , ei+ , eq-, and eq+ are unit vectors of the kDB systems
when the k vectors are pointing to different directions in the chiral
vectors such that:
media, with the same
el
power density,
1(8)1, (Figures 4j to 41). All are shown as a function
of x and z for three contradistinctive cases where 8, P, and ~o
took different values, which were all impedance-matching but not
index-matching situations. One column represents one set of
parameter values, respectively. The first two slabs were chiral
nihility, with
8
=1x 10-5 80
and P = 1x 10-5 Po. The last slab was
not a chiral nihility medium, but had
k:
The
are the x components of wavenumbers for different characteristic waves inside the chiral media:
where the ()± are the angles of refraction for both circularly polarized waves in the chiral slab, stated in the same way as Equation (8). The eight amplitude coefficients for both circularly polarized waves, R±, A±, B±, and T± can be found by matching the
boundary conditions for the tangential (y and z) electric and magnetic fields at the two interfaces of the slab. This generates a linearalgebraic-equation system of eight equations. Under the impedance-matched circumstance 17 =170' all the eight coefficients can be
obtained as follows:
84
)COS()i COS()±
+ (-1 + ei2ki d
j dkzlJf(kz)[ A- (e\ -e2-i )eik;X+ikzz
-00
where
2
)COs ()i -
in this three-layer case.
In a chiral slab,
E=
(20)
I
where
-00
where ktx
+
cos ().cos O:
DE
00
(19)
'
8
= 80 and P = Po.
Since a linearly polarized incident wave is not the
characteristic wave of chiral media - differently from the previous
section with LHCP incidence - a TE incidence is split into two
refracted waves through the chiral slab. When under oblique beam
incidence, at the exit of the slab, two parallel transmitting beams
with opposite circular polarizations are emitted at different locations on the second interface of the chiral slab, as shown in Figure 4. This special case can be a possible application of the slab as
a wave splitter. Besides a slab, if a chiral nihility lens has a curved
surface, it is able to split a linearly polarized Gaussian beam into
right-circularly and left-circularly polarized beams traveling in different directions [16]. In order to obtain high wave-splitting quality
and efficiency of the slab, the values of the constitutive parameters
of the chiral media should be carefully selected. There is a tradeoff
between the distance and the transmitted power of the two circularly polarized wave beams. As in the examples of Figure 4, under
the same thickness, d, of the chiral slab and the same ~o value,
chirality nihility cases are better at separating a beam with different
polarizations, because one of the circularly polarized waves inside
the chirality nihility media is negatively refracted. Although negative refraction also occurs in chiral media that are not chiral nihility
media, it requires that the chiral parameter be large enough, or that
the chirality be combined with an electric plasma [17, 18]. On the
IEEEAntennasand Propagation Magazine, Vol. 51, No.4, August 2009
Figure 5. The time-averaged power density for a Gaussian beam incident upon index-matched chiral nihility slabs with angles
of incidence of (a) 30 0, (b) 100, and (c) 00. The units for the time-averaged power density shown are mW / m 2 •
IEEE Antennas and Propagation Magazine, Vol. 51, No.4, August 2009
85
other
hand,
comparing
the
two
chiral
nihility
examples
( e = 1 x 10-5 &0, J.1 = 1x 10-5 J.10)' we can see that the smaller chirality value (;0 = 0.5 ) results in a larger distance of the two outward
beams, but results in a lower transmission power, due to the greater
mismatch in refraction index, n±::: ±~o' for both of the two
characteristic waves. As was the situation for Figures 2b and 2d,
the reflection amplitudes were small compared with the transmission amplitudes in Figures 4j to 41. Similarly, we can still find the
reflected power through the trails in the interference patterns on the
air side near the half-space boundary.
Figure 5 shows the situation with the chiral nihility slab having an index matched (~o = ±1) to free space. The incident beam
was totally transmitted and divided into two different polarizations
on the other side of the slab. Figure 5a shows the case with the
incident angle equal to 30°. Near the left interface of the slab, the
two polarized transmitted beams were close to each other; interference patterns of these two beams can be clearly observed. After
propagating a distance along the slab, these two beams were well
separated: the structures can thus be used as a beam splitter. For
smaller incident angles, as shown in Figures 5b and 5c, the two
beams were not well apart inside the slab. They recombined into a
linear polarization that is rotated when compared to the incident
polarization, which is the famous optical-activity effect. According
to Equation (28) of [7], when the slab is chiral nihility, the rotation
angle becomes an equation independent of e and J.1 :
E)OA
= 27fd~0 cos()±.
The only side effect is that the recovered
beam in free space is widened, as shown in Figure 5b with a 10°
incidence. However, Figure 5c shows this beam widening will not
happen under normal incidence, and a wave of "standing phase"
[8] is produced inside the chiral nihility slab.
4. Conclusion
The problem of a tapered beam, constructed by a superposition of plane waves with a Gaussian amplitude spectrum, incident
upon a matched chiral half space and a matched chiral slab, has
been solved. The special case where the chiral media are chiral
nihility media in both situations was discussed. An illustration of
the propagation of a Gaussian beam was provided, with different
chirality values. It was shown that under incidences with different
polarizations, chiral nihility media that are matched to free space
exhibit different wave propagating characteristics, depending on
the chirality values. Such behavior can be used in lots of applications, such as beam splitting, beam widening, imaging, etc.
5. Acknowledgment
The authors are grateful to Prof. Joseph R. Mautz for discussions on waves in chiral media. This work was sponsored by the
Chinese National Science Foundation, under grants Nos. 60801005
and 60531020; in part by the NCET-07-0750; the ZJNSF
(RI080320); the PhD Programs Foundation of MEC (No.
20070335120); the ONR under Contract No. N00014-06-1-0001;
and the Department of the Air Force under Air Force Contract No.
FI9628-00-C-0002.
86
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194101, November 2007. ~lV
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