Confirmation of Cylindrical Perfect Invisibility Cloak Using
  Fourier-Bessel Analysis
Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations
Zhichao Ruan∗1,2, Min Yan∗1, Curtis W. Neff1, and Min Qiu1†
1Laboratory of Optics, Photonics and Quantum Electronics,
Department of Microelectronics and Applied Physics,
Royal Institute of Technology (KTH),
Electrum 229, 16440 Kista, Sweden
2Joint Research Center of Photonics of the Royal
Institute of Technology (Sweden) and Zhejiang University,
Zhejiang University, Yu-Quan, 310027 Hangzhou, PR China
Abstract
A cylindrical wave expansion method is developed to obtain the scattering field for an ideal
two-dimensional cylindrical invisibility cloak. A near-ideal model of the invisibility cloak is set up
to solve the boundary problem at the inner boundary of the cloak shell. We confirm that a cloak
with the ideal material parameters is a perfect invisibility cloak by systematically studying the
change of the scattering coefficients from the near-ideal case to the ideal one. However, due to the
slow convergence of the zeroth order scattering coefficients, a tiny perturbation on the cloak would
induce a noticeable field scattering and penetration.
∗ These authors contributed equally to this work.
† Corresponding author. Electronic address: min@kth.se.
http://arxiv.org/abs/0704.1183v2
The exciting issue of exotic materials invisible to electromagnetic (EM) waves was dis-
cussed in recent works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Based on a coordinate transformation
of Maxwell’s equations, Pendry et al. first proposed an invisibility cloak, which can protect
objects inside the cloak from detection [1]: When EM waves pass through the invisibility
cloak, the cloak will deflect the waves, guide them around the object, and return them to
the original propagation direction without perturbing the exterior field. Numerical methods
have been applied to solve the EM problem involving invisibility cloaks [6, 9], and an experi-
mental result of the invisibility cloak using metamaterial with simplified material parameters
has also recently been reported [7]. Yet, the ideal invisibility cloak has not been confirmed
as a perfect cloak, due to the extreme material parameters required (zero or infinity) in the
ideal cloak when approaching the inner boundary. Also, numerical methods usually describe
the material parameters discretely, which can be computationally intensive in extreme cases.
Thus it is preferable to use an analytical or semi-analytical method whenever possible.
In this paper, we will study the scattering for an ideal invisibility cloak. We focus
our analysis on the 2D cylindrical cloak, because the wave equation can be simplified in
comparison with the 3D case, and a 2D invisibility cloak is more feasible to fabricate [7].
Here we take advantage of the cylindrical geometry of the structure and use the cylindrical
wave expansion method to study the device semi-analytically. To avoid extreme values
(zeros or infinity) of material parameters at the cloak’s inner surface, we introduce a small
perturbation into the ideal cloak, and allow the perturbation to approach zero to study the
scattering problem for the ideal cloak. Such an asymptotic analysis not only can confirm
whether the ideal cloak would be perfectly invisible or not, it also provides hints on how
sensitive such a device is to finite perturbations. A sensitivity analysis of the invisibility cloak
directly determines the possibility of its application. Our studies show that the cylindrical
invisibility cloak is very sensitive to tiny perturbations of the material parameters.
First, let’s look at the wave equation inside a cylindrical cloak. According to Ref. [1], a
simple transformation
r′ = b−a
r + a, θ′ = θ, z′ = z (1)
can compress space from the cylindrical region 0 < r < b into the annular region a < r′ < b,
where a is the inner radius of the cloak, b is the outer radius of the cloak, and r, θ and z
(r′, θ′ and z′) are the radial, angular and vertical coordinates in the original (transformed)
system, respectively. Following the approach in Ref. [1], the permittivity and permeability
tensor components for the cloak shell can be given as
εr = µr =
, εθ = µθ =
εz = µz =
)2 r−a
and air is assumed for the ambient environment and the interior regions. In the following,
the transverse-electric (TE) polarized electromagnetic field is considered (i.e. the electrical
field only exists in the z-direction) , however the transverse-magnetic derivation follows in
similar manner. Throughout the paper, a exp(−iωt) time dependence is assumed. For the
TE-polarized wave, only εz, µr, and µθ are relevant to the following general wave equation
governing the Ez field in the cloak’s cylindrical coordinate
) + k20Ez = 0, (3)
where k0 is the wave vector of light in vacuum. If we substitute Eq. 2 for εz, µr, and µθ, we
+ rµθ
+ εzµθr
2k20Ez +
= 0 (4)
Equation 4can be solved by a separation of variables Ez = Ψ(r)Θ(θ) and the introduction
of a constant l:
(r − a)2
+ (r − a)
)2(r − a)2k20 − l
2]Ψ = 0 (5)
+ l2Θ = 0. (6)
Equation 5 is the lth-order Bessel differential equation, and the general solution of Eq. 6 is
exp(ilθ). Therefore, there exists a simple set of solutions to Ez in the cloak shell of the form
Fl(k1(r − a)) exp(ilθ) (7)
where k1 = k0b/(b − a), Fl is the l-order Bessel function, and l is an integer number as
required by the rotational boundary condition.
Let us consider the scattering problem in which an arbitrary wave is incident on the
cloak. According to the rigorous scattering theory [12], the incident field in the 2D case can
be expanded in the cloak’s coordinates with the following expression
Einz =
αinl Jl(k0r) exp(ilθ), (8)
where Jl is the l
th-order Bessel function of the first kind. The scattering field can also be
expanded as
Escz =
αscl Hl(k0r) exp(ilθ), (9)
where Hl is the l
th-order Hankel function of the first kind.
FIG. 1: The schematic of a near-ideal invisibility cloak: The distribution of the material parameter
is the same as the ideal one shown in Eq. 2, and the outer boundary is still fixed at r = b. However,
the actual inner boundary is at r = a+ δ, where δ is a very small positive number.
We note that the scattering coefficients cannot be directly obtained for the ideal cloak
since εz → 0, µr → 0, and µθ → ∞ when r → a, and the Bessel function of the second
kind in Eq. 7 has a singularity at r = a. In order to circumvent this, we introduce a small
perturbation to the ideal cloak which we refer to as the near-ideal cloak, see Fig. 1. We
expand the inner boundary of the cloak shell slightly, so that it is located at r = a+δ, where
δ is a very small positive number. However, the material parameters are still calculated
according to Eq. 2 as if the inner boundary is unchanged. The outer boundary remains
fixed at r = b. When δ → 0, our model will be equivalent to the ideal cloak. Now the
electric-field in each region can be given by
(b < r) Ez =
αinl Jl(k0r) exp(ilθ) + α
l Hl(k0r) exp(ilθ)
(a+ δ < r < b) Ez =
α1l Jl(k1(r − a)) exp(ilθ) + α
lHl(k1(r − a)) exp(ilθ)
(r < a+ δ) Ez =
α3l Jl(k0r) exp(ilθ)
where αil(i = 1, 2, 3) are the expansion coefficients for the resulting field inside the cloak.
The tangential fields Ez and Hθ (which can be obtained from Ez), should be continuous
across the interfaces at r = a+ δ and r = b; and the orthogonality of exp(ilθ) allows waves
in each Bessel order to decouple. Thus, we can have the following four equations:
αinl Jl(k0b) + α
l Hl(k0b) = α
l Jl(k1(b− a)) + α
lHl(k1(b− a)) (11a)
α1l Jl(k1δ) + α
lHl(k1δ) = α
l Jl(k0(a+ δ)) (11b)
l (k0b) + k0α
l(k0b) =
µθ(b)
α1l J
l (k1(b− a)) +
µθ(b)
l(k1(b− a)) (11c)
µθ(a+ δ)
α1l J
l (k1δ) +
µθ(a+ δ)
l(k1δ) = k0α
l (k0(a+ δ)) (11d)
which is a set of linear equations. Thus each order expansion coefficient in each material
region can be exactly solved. In turn we can obtain the fields in each region.
As a direct result of this set linear equations, we can prove that when δ → 0, αscl = α
l → 0
, α1l = α
l , and α
l → 0 for any α
l , i.e., the ideal cloak is a perfect invisibility cloak. Firstly, it
can be assumed that |αil(i = sc, 1, 2, 3)| must be finite. Otherwise, the scattering field would
be infinite if the incident field has the lth order component. Secondly, due to k1(b−a) = k0b
and k1 = k0µθ(b), when δ → 0, Eq. 11a and 11c become (α
l )Jl(k0b)+(α
l )Hl(k0b) =
0 and (αinl − α
l (k0b) + (α
l − α
l(k0b) = 0, respectively. Since b can be arbitrary and
the Bessel functions are not always zeros, αinl = α
l and α
l = α
l must be satisfied. Thirdly,
from Eq. 11b, we can obtain the following inequality
∣α2lHl(k1δ)
∣α3l Jl(k0(a+ δ))
∣α1l Jl(k1δ)
∣ . (12)
When δ → 0, the right side of the above inequality approaches a finite value but |Hl(k1δ)|
approaches infinity on the left side. Thus, |α2l | must approach zero. Finally, from Eq. 11b,
we can also obtain that |α2lH
l(k1δ)| ≤
J ′l (k0(a+ δ))
+|α1l J
l (k1δ)|. While from Eq. 11d,
we have
l(k0(a+ δ))
µθ(a+ δ)
α1l J
l(k1δ)
µθ(a+ δ)
l(k1δ)
. (13)
Since µθ(a+ δ) → ∞ and the right side of the above inequality approaches zero when δ → 0,
we obtain that |α3l | → 0. Consequently, this argument proves that the scattering field and
the field in the interior region of the cloak are zero when δ = 0, i.e., the ideal cloak is a
perfect invisibility cloak.
FIG. 2: (Color online) (a) Snapshot of the resulting electric-field distribution, (b) the corresponding
norm in the vicinity of the cloaked object, and (c) the snapshot of the corresponding scattering field
outside the cloak for the near-ideal cloak with δ = 10−5a and when a plane wave is perpendicularly
incident on the cloak. The black lines outline the cloak shell. Axis unit: meter.
Although we have just confirmed that the ideal cloak can provide perfect invisibility,
further study the near-ideal cloak by the above analytical method illuminates how sensitive
the parameter δ is to the performance of the cloak. As an example, we use the same material
parameters in Ref. [6] where the inner radius of the cloak is a = 0.1m, the outer radius of
the cloak is b = 0.2m, and the frequency of the incident plane wave is 2GHz. Similarly, we
also consider a plane wave incident on the cloak, where the expansion coefficients in Eq. 8
αinl = i
lA exp(−ik0r1 cos(ϕ+ θ1)− ilϕ), (14)
where (r1, θ1, 0) is the coordinate of the phase reference point, A is the amplitude of the
plane wave, and ϕ is the incident angle [13]. Here the phase reference point is set at r1 = 4a
and θ1 = π, the amplitude is A = 1, and the incident angle is ϕ = 0 (i.e. the plane wave
propagates from left to right). We use 31 Bessel terms (−15 ≤ l ≤ 15) to calculate the
scattering field for the near-ideal cloak with δ = 10−5a. The number of expansion terms
is sufficient for convergence of the calculated fields. Figure 2 shows the snapshot of the
resulting electric-field distribution (i.e. the real part of the electric-field phasor), and the
corresponding norm in the vicinity of the cloaked object. The electric-field distribution
clearly demonstrates the cloaking effect of the near-ideal cloak to the incident plane wave.
However, the norm of the electric-field (Fig. 2 (b)) reveals that there is still a little bit of the
field in the cloak interior and an obvious scattering ripple around the cloak. The amplitude
of the resulting electric-field at the center is 0.197. The snapshot of the scattering field
(Fig. 2 (c)) shows that it propagates almost isotropically in all angles. Even though the
amplitude of the scattering field is much smaller than that of the incident plane wave, the
interference of the incident plane wave and the scattering field creates the ripples in the
norm (Fig. 2 (b)).
Since each order expansion coefficient of the scattering field is only relevant to each order
expansion coefficient of the incident field (cf. Eq. 11), we can define the scattering coefficient
for each order as
cscl =
. (15)
These coefficients for the field inside the cloak c
= αil/α
l , i = 1, 2, 3 can also be defined
in the same way. To study the ideal cloak, we more δ closer to 0. The amplitude and the
phase of these coefficients for 10−8a < δ < 10−2a are shown in Fig. 3, where (a)-(b) and
(c)-(d) correspond to the cases of l = 0 and l = 1 , respectively.
From Fig. 3, it is clear that c
is always equal to 1 for both cases. That is, the incident
field propagates into the cloak without any reflection at the outer boundary, which coincides
with the explanation of the cloaking effect from the coordination transformation approach
FIG. 3: (Color online) The amplitude and phase of the scattering coefficients for the different δ,
where (a)-(b) and (c)-(d) correspond to the cases of l = 0 and l = 1 respectively. csc
is denoted
by the blue point-dashed line, c
(the solid black line), c
(the red circle-marked), and c
green star-marked).
[1]. The same behavior for the scattering fields occurs at the outer boundary, where they
propagate from inside to outside without any reflection, thus cscl is always equal to c
Our computational results also confirm that both cscl and c
approach zero when δ → 0.
In particular, compared with the case of l = 0, |cscl | and
for l = 1 are much smaller,
and approach zero more rapidly. This is also observed for the other higher order cases.
Thus, in the case of the plane wave incident, where |αinl | is the same for each order, the
dominating term of the scattering field outside of the cloak is of the form of the zeroth-order
Hankel function of the first kind. Meanwhile, the resulting field in the interior region has a
dominating term of J0(k0r) . This explains the near azimuthally invariable distribution of
the field in the interior region (see Fig. 2(a)) and the scattering field outside the cloak (see
Fig. 2(c)), which has also been mentioned in Ref. [6].
It is worth noting that the zeroth order scattering coefficients csc0 and c
0 decrease ex-
tremely slowly with reduced δ, e.g. when δ is decreased from 10−5a to 10−8a, |csc0 | decreased
only from 0.175 to 0.099. By utilizing the arbitrary calculation precision of the software
MATHEMATICA, we found that the convergence of the limit is so slow that even for
δ = 10−99a (i.e. εz ≈ 4 × 10
−99, µr ≈ 10
−99, and µθ ≈ 10
99 at the inner boundary in this
case), |csc0 | = 6.973 × 10
−3. Therefore, we conclude that even though an cloak with the
ideal material parameters in Ref. [1] is a perfect cloak, a non-ideal invisibility cloak does not
provide a good enough cloaking effect due to the slow convergence of |csc0 | and
In conclusion, we have used the cylindrical wave expansion method to study the electro-
magnetic scattering properties of a 2D invisibility cloak. A near-ideal model of the invisibility
cloak is set up to solve the boundary problem at the inner boundary of the cloak shell. By
systematically studying the change of the scattering coefficients from the near-ideal case
to the ideal one, we confirm that the cloak with the ideal material parameter is a perfect
invisibility cloak. But due to the slow convergence of the scattering coefficients, a tiny per-
turbation on the cloak would induce a noticeable field scattering and penetration. We also
proved that the scattered and penetrated fields are dominated by zeroth-order cylindrical
waves. Though our work has focused on the 2D cylindrical cloak, it can be reliably ex-
tended to the 3D spherical case. Our method and results are also useful for either designing
or detecting this type of the invisibility cloak.
This work is supported by the Swedish Foundation for Strategic Research (SSF) through
the INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish
Research Council (VR). Z.C.R. acknowledges the partial support from the National Basic
Research Program (973) of China under Project No. 2004CB719800.
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