arXiv:0706.0655v2 [physics.optics] 25 Feb 2008
Cylindrical Invisibility Cloak with Simplified Material Parameters
is Inherently Visible
Min Yan, Zhichao Ruan, and Min Qiu∗
Department of Microelectronics and Applied Physics,
Royal Institute of Technology, Electrum 229, 16440 Kista, Sweden
(Dated: February 25, 2008)
It was proposed that perfect invisibility cloaks can be constructed for hiding objects from electro-
magnetic illumination (Pendry et al., Science 312, p. 1780). The cylindrical cloaks experimentally
demonstrated (Schurig et al., Science 314, p. 997) and theoretically proposed (Cai et al., Nat. Pho-
ton. 1, p. 224) have however simplified material parameters in order to facilitate easier realization
as well as to avoid infinities in optical constants. Here we show that the cylindrical cloaks with
simplified material parameters inherently allow the zeroth-order cylindrical wave to pass through
the cloak as if the cloak is made of a homogeneous isotropic medium, and thus visible. To all
high-order cylindrical waves, our numerical simulation suggests that the simplified cloak inherits
some properties of the ideal cloak, but finite scatterings exist.
PACS numbers: 41.20.-q, 42.25.Bs, 42.79.Wc
∗Electronic address: firstname.lastname@example.org
Recently there has been an increase of interest in realizing invisibility cloaks [1, 2, 3, 4,
5, 6, 7, 8]. In particular, the coordinate transformation method proposed in  is noticed
to be especially powerful for designing invisibility cloaks that can in principle completely
shield enclosed objects from electromagnetic (EM) illumination and at the same time cause
zero disturbance to the foreign EM field. In 2D case, the cylindrical cloak obtained through
such technique has anisotropic, spatially varying optical constants. In addition, some of the
material parameters have infinite values at the interior surface of the cloak. To facilitate
experimental realization at microwave frequencies, Schurig et al. have simplified the material
properties such that only one material parameter is gradient and the requirement on infinite
material constant is lifted . The authors claimed that, in comparison to the ideal cloak, the
simplified cloak maintains the power-flow bending with the penalty of nonzero reflectance
at the outer interface. Similar simplification on a cylindrical cloak at optical frequencies
is also employed in Ref. . In this paper we provide a systematic theoretical study on
simplified cylindrical cloaks. It is found that the bare device constructed using the simplified
medium fails to be invisible. In addition, the device does not possess a spatial region that
is completely in isolation from the outside world electromagnetically. Hence, perfect hiding
with such a simplfied cloak is not possible.
Consider a cylindrical cloak with its inner and outer boudaries positioned at r = a and r =
b respectively. Both domains inside and outside the cloak are air. The structure is in general
a three-layered cylindrical scatterer. We refer to the layers from inside to outside as layer
1, 2 and 3, respectively. Our analysis, similar to all previous works, is on normal incidence.
That is, the k vector is perpendicular to the cloak cylinder axis. The EM wave is assumed
to have TE polarization (i.e. electric field only exists in z direction). The TM polarization
case can be derived by making E → −H, ε → µ, and µ → ε substitutions. By default,
we choose the cloak’s cylindrical coordinate as the global coordinate. In a homogeneous
material region (i.e. cloak interior and exterior) the general solution is expressible in Bessel
functions. Within the cloak medium, the general wave equation that governs the Ezfield
can be written as
0εzEz= 0, (1)
where k0is the free-space wave number, µθ, µrand εrare polarization-dependent perme-
ability/permitivity profiles of the cloak. The time dependence exp(iωt) has been used for
deriving Eq. 1.
Ideally, the cloak is designed to compress all fields within a cylindrical air region r < b
into the cylindrical annular region a < r < b. A corresponding coordinate transformation
leads to a set of anisotropic and spatially variant material constants in the cloak shell, as
described in . Such an annular cylinder can indeed provide perfect invisibility cloaking
, but it requires infinite values of optical constants at the cloak’s inner boundary. To
circumvent the fabrication difficulty, the simplified parameters were used . They are in
the form of
To achieve cloaking for TM waves, the same set of material parameters, but for εr, εθand
µz, is used . However, we notice that the precedure for simplification of the material
parameters adopted in  is questionable, as the derivation has assumed beforehand that µθ
is a constant. It is obvious that the invariant µθcan be taken out of the differential operator
in Eq. 1. Therefore wave behavior within the cloak shell is altered as compared to that in
an ideal cloak.
Since the material parameters in Eq. 2 are azimuthally invariant (which is also true for
the ideal parameter set), we can use the variable separation Ez= Ψ(r)Θ(θ). Eq. 1 can then
be decomposed into
dθ2+ m2Θ = 0,
0rεzΨ − m21
rµrΨ = 0, (4)
where m is an integer. The solution to Eq. 3 is exp(imθ). Equation 4 is a second-order ho-
mogeneous differential equation. Two independent solutions are expected. At this moment,
we assume the solution to Eq. 4, for a fixed m, can be written in general as AmQm+BmRm,
where Amand Bmare constants. Qmand Rmare functions of r. Now valid field solutions
in three layers (denoted by superscripts) can be described as
mJm(k0r) + B3
drical wave. The Jmand H(2)
m is the Hankel function of the second kind, which represents outward-travelling cylin-
m terms in the 3rd layer are physically in correspondence to the
incident and scattered waves, respectively. Hence, the scattering problem becomes to solve
for, most importantly, A1
m(transmitted field) and B3
m(scattered field) subject to a given in-
m. Comparatively A2
mare physically less interesting. The coefficients
are solved by matching the tangential fields (Ezand Hθ) at the layer interfaces. Due to the
orthogonality of the function exp(imθ), the cylindrical waves in different orders decouple.
We hence can examine the transmission and scattering of the cloak for each individual order
By substituting the simplified material parameters into Eq. 4, we obtain
(r − a)2d2Ψ
dr2+(r − a)2
(r − a)2
b − a
Ψ = 0. (8)
Equation 8 has two non-essential singularities at r = 0 and r = a for m ?= 0. It is worthwhile
to mention that, with the ideal parameters, Eq. 4 can be written instead as
(r − a)2d2Ψ
dr2+ (r − a)dΨ
(r − a)2
b − a
Ψ = 0.(9)
When m = 0, Eq 8 can be further simplified to
b − a
Ψ = 0. (10)
This is the zeroth-order Bessel differential equation. Its non-essential singularity remains at
r = 0. Equation 10 suggests that an incoming zeroth-order cylindrical wave would effectively
see the simplified cloak as a homogeneous isotropic medium whose effective refractive index
b−a. Its transmission through the cloak shell is therefore determined by the etalon
effect of the finite medium.
When m ?= 0, the wave solution is governed by Eq. 8. By comparing Eqs. 9 and 8, we
see that at radial positions r >> a, Eq. 8 asympototically resembles Eq. 9 due to r−a ≈ r.
This indicates that all high-order cylindrical waves tend to behave similarly in both media
at r >> a positions, and hence the importance of parameter b, at which the cloak medium is
truncated. In the following, we will derive the scattering coefficient smsubject to individual
cylindrical wave incidences. The scattering coefficient in each cylindrical order is defined as
m|. Analytic derivation of smcan be done if two solutions to Eq. 8 (i.e. Qm
and Rm) are known in closed form. However, despite the analogueness to Eq. 9, Eq. 8 fails
the analytic Frobenius method . Here we tackle the problem through the finite-element
method. The field outside the cloak is computed numerically, and then is used for deriving
mthrough a fitting procedure. smis known in turn. Solutions with
different azimuthal orders are obtained by varying the azimuthal dependence of a circular
current source outside the cloak. The scattering problem is numerically manageable since
functionals εzµθ and
µrin Eq. 4, unlike in the ideal cloak case, are both finite and do
not possess any removable singularity for the simplified medium. The commercial software
COMSOL is deployed to carry out calculations. For our case study, we fix a = 0.1m and
operating frequency f = 2GHz. The performance of the cloak is examined as b is increased
from 0.2m. Similar parameters are also found in .
0.20.250.3 0.350.4 0.45 0.5 0.550.6
FIG. 1: Variation of the scattering coefficients, examined in each cylindrical wave order, as a
function of b. For m = 1,2, the curves are fitted using Savitzky-Golay smoothing filter according
to numerically derived data points (dots).
The scattering coefficients in different azimuthal orders as a function of b are shown in Fig.
1. As expected, the zeroth-order scattering coefficient is quite distinct from others due to the
different governing wave equation. In fact, analytic solution exists when m = 0, as field in the
cloak medium are Bessel functions . The excellent agreement between the numerical and
analytic results for zeroth-order scattering coefficient confirms the validity and accuracy of
our approach. Using the analytic technique, the zeroth-order scattering coefficient is found to
converge to 0.867 with respect to b. Despite that the effective index of the cloak approaches
to 1 as b increases, the phase variation of the zeroth-order wave within the cloak medium is
increasing, as k0
b−a(b−a) = k0b. This explains why the scattering coefficient converges to a
value other than 0. The existence of zeroth-order scattering coefficient effectively disqualifies
the cloak to be completely invisible.
Compared to the zeroth-order scattering coefficient, the high-order scattering coefficients
(only those for m = 1,2 are shown in Fig. 1) are noticed to be in a similar oscillatory
fashion, and in general much smaller. The scattering coefficient tends to converge to a value
closer to zero when the order number m increases. Over certain ranges of b value (e.g.
around b = 0.225m for m = 1), the computed B3
mchanges sign and hence the resulted
scattering coefficient seems to be flipped from a negative value. We should attribute the
relatively small high-order scattering coeffcients to the cloak’s partial inherence of the ideal
cloak based on coordinate transformation . However, our numerical result shows that
the high-order scattering coefficients do not converge to zero even when the cloak wall is
Besides the requirement of zero scattering (invisibility), a device also need to possess
a spatial region which is in complete EM isolation from the exterior world in order to be
an invisibility cloak. Therefore it is meaningful to know how much field penetrates into
the simplified cloak subject to a foreign EM illumination. Again, the problem is studied
by examining the individual cylindrical wave components separately.The transmission
coefficient, defined as tm= |A1
m|, is used to characterize the field transmission. When
m = 0, the amount of field transmitted into the cloak interior can be analytically derived,
which is shown in Fig. 2 as a function of b. The transmission is noticed to be oscillatory, and
converging to 1 as b increases. Numerical calculation is also superimposed for validation.
When m ?= 0, the FEM calculations show that the field inside the cloak is almost zero.
The corresponding transmission coefficients are exclusively smaller than 0.005, hence are
not plotted in Fig. 2. This indicates that the contour r = a provides an insulation between
its enclosed domain and the exterior domain, but only for all high-order cylindrical waves.
Therefore, any objects placed inside the cloak are exposed to the zeroth-order cylindrical wave
component. Reversely, the zeroth-order wave component of an EM source placed within the
cloak (or scattered wave by objects inside the cloak) will transmit out. As a result, objects
enclosed by a simplified cloak is sensible by a foreign detection unit.
Next we numerically demonstrate (in COMSOL) scattering by the simplified cloaks with
a plane-wave incidence. The incident plane wave travels from left to right and has the
amplitude of 1. When b = 0.2m, the Ezsnapshot, Eznorm, and the scattered Ezsnapshot
are plotted in Fig. 3(a1)-(a3), respectively. It is noticed that the amplitude of the scattered
field is about one half of the incident field. From the scattered field distribution, high-order
Bessel terms constitute a significant portion. Scattering aside, the field inside the cloak
shell is seen to have only zeroth-order Bessel term. We then increase b to 0.5m, and the
corresponding numerical results are plotted in Fig. 3(b1)-(b3). Compared to the previous
case, scattered field is reduced roughly by half in amplitude, and is now dominated by the
zeroth-order Bessel term. The Eznorm is closer to be uniform outside the cloak, indicating
better invisibility. The overall smaller scattering as well as the dominance of the zeroth-order
scattering for the second cloak agree well with our derviation of the scattering coefficients
in Fig. 1. When b is changed from 0.2m to 0.5m, the transmitted Ezfield at the center of
the cloak increases from 0.6032 to 0.8855 in norm, also in agreement with Fig. 2.
0.1 0.20.3 0.4 0.50.6 0.7 0.8 0.91.0
FIG. 2: The zeroth-order transmission coefficient.
(a1) (a2) (a3)
FIG. 3: (a1)-(a3) Ezsnapshot, Eznorm, and scattered Ezfield, respectively, for a simplified cloak
with a = 0.1m and b = 0.2m. (b1)-(b3) Same fields but for a simplified cloak with a = 0.1m and
b = 0.5m. (c1)-(c3) Same fields but for a near-to-ideal cloak with a = 0.101m and b = 0.5m.
For completeness, we also show the scattering by a near-to-ideal cloak in Fig. 3(c1)-(c3).
The material parameters of the cloak are defined by µr=r−a
r−a, and εz=?
with a = 0.1m and b = 0.5m. To avoid the r = 0.1m critical contour, we let the inner
boundary of the cloak be positioned at r = 0.101m. In  we have confirmed analytically
that the performance of an ideal cloak is extremely sensitive to the position of the cloak’s
inner surface. In particular, the zeroth-order cylindrical wave will experience considerable
reflection and transmission as it meets the cloak shell. This is confirmed in Fig. 3(c1)-(c3).
The scattered field is almost purely the zeroth-order cylindrical wave. Even at such a small
mis-location of the inner boundary, the amount of field leaking into the cloak is handsome,
valued at 0.5466 in norm. By comparing panels (c1)-(c3) and (b1)-(b3) in Fig. 3, we see that
the improved cloak made from simplified medium inherits some merits of the near-to-ideal
cloak, but with overall higher scattering, especially in the monopole wave component.
In conclusion, we have theoretically studied the cylindrical cloaks with simplified material
parameters. Such a simplified cloak is shown to inherit some properties of an ideal cloak
based on coordinate transformation of Maxwell equations. However the penalty of using the
simplified cloak is more than just nonzero reflectance at the cloak boundary. The monopole
component of the incoming wave will always experience relatively high scattering. High-
order cylindrical waves also experience finite (although smaller) scattering even when the
cloak’s wall is kept very thick, as suggested by our numerical simulation. Besides that
the device itself is visible, it cannot completely shield EM field, due to penetration of the
monopole field component. Hence cloaking of objects will not be perfect. Lastly, considering
that the monopole field treats a simplified cloak as a conventional glass tube, detection of
any object placed within the cloak can be greatly enhanced by using a EM source with the
maximum energy on its monopole component.
Acknowledgement: This work is supported by the Swedish Foundation for Strategic Re-
search (SSF) through the INGVAR program, the SSF Strategic Research Center in Photon-
ics, and the Swedish Research Council (VR).
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 G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970), chap. 9, 2nd ed.
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 For comparison purpose, it should be mentioned that the scattering coefficient in any order
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geometry is fixed) or its outer radius (as material and inner radius are fixed).