Download Isolated object invisibility cloak in the visible light

Broadband polygonal invisibility cloak for visible
light
Hongsheng Chen1,2* and Bin Zheng1,2
1
The Electromagnetics Academy at Zhejiang University, Zhejiang University,
Hangzhou 310027, China
2
Department of Information Science & Electronic Engineering, Zhejiang University,
Hangzhou 310027, China
* Corresponding author: [email protected]
1. Polygonal cloak design based on transformation optics
Figure S1 | Illustration of the initial cylindrical cloak design and the proposed general
m-sided polygonal cloak design based on transformation optics. In the initial cylindrical
cloak design3,18, a cylinder in a virtual coordinate space (a) filled with isotropic materials is
transformed to an annulus in the physical coordinate space (b) with inhomogeneous
anisotropic materials. In the general m-sided polygonal cloak design, the virtual coordinate
space between the two polygons is divided by lots of triangular segments (c). A linearly
homogeneous transformation method is applied in each segment along its local coordinate
axes to form a polygonal cloak in the physical coordinate space (d). The hidden region defined
by the center polygon, with radius r1 in the physical coordinate space, is mapped from a much
smaller polygon, with radius r0 in the virtual coordinate space, while the outer polygon with
radius r2 in the physical coordinate space remains the same as that in the virtual coordinate
space.
To better illustrate a general m-sided polygonal cloak design, we first briefly
review the initial cylindrical cloak design3,18 and see how we can make a
column-shaped cloak in visible light. The cylindrical cloak device with coordinate
transformation is illustrated in Fig. S1a. The straight Cartesian coordinates in a virtual
coordinate space (Fig. S1a) filled with isotropic materials of permittivity  and
permeability  (  =1) is transformed to a physical coordinate space (Fig. S1b) with
curved coordinate lines. The transformation can be mathematically defined by3,18:
r '  r1 
r2  r1
r ,  '  , z' z .
r2
(S1)
This transformation leads to the following parallel polarized wave related constitutive
parameters of the cloak region3,18:
2
 r  r ' r1
r ' r1
r'
 '  
,  '  
, z '   2 
.
r'
r ' r1
 r2  r1  r '
(S2)
In this transformation, two limitations may be introduced in the realization of a
macroscopic cloak for visible light. The first one is the extreme values of the material
properties at the inner boundary of the cloak defined as r '  r1 . It results from the
transformation of a point in the virtual coordinate space to a circle with finite size.
This limitation can be conquered, if a small amount of scattering is acceptable, by
simplifying the constitutive parameters or by changing the infinitesimal point to a
circle with finite small size in the virtual coordinate space. This is based on the
common sense that if the scattering of the rays on the object is small, invisibility to
the naked eye is still possible. The second one is the intrinsic inhomogeneity and
anisotropy of the materials resulting from the inhomogeneous transformation. As the
transformation in equation (S1) keeps the angle in the virtual coordinate space and
that in the physical coordinate space to be the same, i.e.  '   , lines closer to the
outer boundary of the cloak are less stretched along the angular direction than those
closer to the inner boundary. This inhomogeneous transformation leads the materials’
constitutive parameter tensors to be spatially dependent, which can be difficult to find
in natural material, or precisely produce using the nano-fabrication process.
To circumvent these two limitations, an m-sided polygonal cloak is proposed. The
circle in the original space in Fig. S1a is changed to an m-sided polygon in Fig. S1c.
In its center is a smaller polygon rotated at an angle of  / m compared to the outer
polygon. The space between the two polygons is divided by several triangular
segments. According to their geometries, the triangular segments in the cloak can be
grouped into two types: Segment I and Segment II with their local coordinates
( ua , va , wa ) and ( ub , vb , wb ), respectively. The following linear homogeneous
transformation is applied in all segments along their local coordinate axes:
ua '  ua /  u , va '   v va , wa '  wa , for Segment I, and
ub '  ub /  , vb '  vb , wb '  wb , for Segment II,
where   (r2 cos

m
 r0 ) (r2 cos

m
 r1 ) ,  u  (r2  r0 cos
(S3)

m
) (r2  r1 cos

m
) , and
 v  r1 r0 are the compression or extension ratios of the space. Through this
transformation, we get the following parallel polarized wave related constitutive
parameters of the cloak:
 uI '   /( u v ) ,  vI '   u v , wI '   u /  v , for Segment I, and
 uII '   /  ,  vII '   ,  wII '   , for Segment II.
(S4)
The general m-sided polygonal cloak is therefore composed of lots of triangular
segments with homogenous and anisotropic materials. As far as the trajectory of the
ray is concerned, the constitutive parameters of the cloak can be simplified as a
non-magnetic form:
 uI '   /  v2 ,  vI '   u2 ,  wI '  1 , for Segment I, and
 uII '   ,  vII '   2 ,  wII '  1 , for Segment II.
(S5)
Fig. S1d shows a 20-sided polygonal cloak. We see the trajectories of the rays in the
polygonal cloak (Fig. S1d) become very close to those in the cylindrical cloak (Fig.
S1b). In the main text, we show a hexagonal cloak (m=6) for experimental
demonstration.
II. k-surface diagram to derive the parameters of the simplified
hexagonal cloak
Figure S2 | k-surface diagram to design and interpret the refraction behavior of the rays
in the hexagonal cloaks. a, The inset shows the transformation based hexagonal cloak
composed of twelve triangular segments and the propagation of the ray inside the cloak. The
ray with a Poynting vector S0 along the x direction is incident onto the cloak from a
homogenous background medium (Region 0), whose k-surface is represented by the cyan
circle. Region 1 and Region 3 are both Segment II medium. Their k-surfaces are represented
by blue ellipses with different orientations determined by the orientations of the optical axis.
The ellipse in green is the k-surface of the Segment I medium in Region 2. The k vectors, k0, k1,
k2, k3, and the Poynting vectors, S0, S1, S2, S3, represent the phase propagating vector and the
energy propagating vector of the ray in Region 0, Region 1, Region 2, and Region 3,
respectively. B1, B2, and B3 represent the interfaces shown in the inset. b, The inset shows the
simplified hexagonal cloak composed of only Segment II medium. The Segment I medium is
removed so that B2 coincides with B3. The k-surface diagram shows the refraction behavior of
the ray in the cloak. The k-surface of the background medium (Region 0), which is represented
by the cyan circle, is bigger than that in (a). c, Ray tracing of the light passing through the
transformation cloak. The outer radius is r2=13 mm, the inner radius is r1=1.95 mm. The
background material has a relative permittivity of   1.492 and a relative permeability of
  1 . The Segment I medium has the following electromagnetic parameters:  uI '   / 2.22 ,
 vI '  1.082  ,  wI '  1 . The Segment II medium is the calcite crystal with  uII '  1.492 ,
 vII '  1.662 ,  wII '  1 . d, Ray tracing of the light passing through the simplified cloak. The
outer radius is r2=13 mm, the inner radius is r1=1.95 mm. The background material has a
relative permittivity of   1.722 and a relative permeability of   1 . The Segment II
medium is the calcite crystal with  uII '  1.492 ,  vII '  1.662 ,  wII '  1 . These parameters
are implemented for experimental demonstrations except that the inner radius of the cloak in
the experiment is r1=1.5 mm, which is a bit smaller in order to tolerate some fabrication errors.
The design of the simplified cloak used in the experimental demonstration is
based on the trajectory of the horizontal propagating ray marked in red (Fig. 1b). A
k-surface diagram29 is used to demonstrate how the ray refracts at each interface.
Mathematically, the k-surface of the homogenous background medium (Region 0) is
defined by kx2  k y2   2  , whose curve is represented by the cyan circle in Fig. S2a.
The ray with a wave vector of k0 = xk0 and Poynting vector S0= xS 0 is incident along
the x direction and propagates through the interface (B1) between the background
medium and the first Segment II medium (Region 1). The k vector of the light in
Region 1, k1= uku  vkv , is required to satisfy the following dispersion relation:
ku2

kv2
 '  '
II
v
II
u
  2  wII ' . Its k-surface is therefore characterized by an ellipse in blue with
the optical axis along the v  x cos(5 / 6)  y sin(5 / 6) direction. At the interface,
the ray obeys the Snell’s law, i.e. the tangential wave vector components of the
incident ray (k0B1) and the refracted ray (k1B1) should be equal. The refracted ray
therefore propagates along the S1 direction, which is always perpendicular to the
elliptical surface29, and headed to the second interface (B2) between Region 1 and
Region 2. The k vector of the light in Region 2, k2= uku  vkv , is required to satisfy
the following dispersion relation:
ku2

kv2
 '  '
I
v
I
u
  2 wI ' . Its k surface is therefore
characterized by an ellipse marked in green, with the optical axis along the
v  x cos(2 / 3)  y sin(2 / 3) direction (Fig. S2a). The refraction law requires the
tangential wave vectors of the incident ray (k1B2) and the refracted ray (k2B2) at the
two sides of the interface to be the same, leading to the refracted ray propagating
along the S2 direction. At the third interface (B3) the ray is refracted to the second
Segment II medium (Region 3), whose k-surface is an ellipse marked in blue, the
same as that in Region 1, but with an optical axis along the y direction. As
refraction law requires k2B3= k3B3, we find the refracted ray in the second Segment
II medium is along the x direction. From the k-surface diagram, we see that k3 and
S3 coincide with k0 and S0, respectively. Due to the symmetric pattern and the
reciprocal principle, the ray will propagate through the remaining part of the cloak
and appear at the background medium without any deviation. The hidden object is
therefore invisible to this ray.
Now let’s see what we can get if we rotate the interfaces B2 and B3 in the
following way: B2 moves in an anti-clockwise direction according to the top vertex of
the isosceles triangle of the Segment I medium in Region 2, while the interface B3
moves in a clockwise direction. The same rotation is applied for the boundaries at the
other regions filled with Segment I medium. When B2 and B3 both coincide with the
symmetric line of the triangle of Region 2, Region 2 disappears. As we apply the
same rotation to the boundaries at all the regions filled with Segment I medium, all
the Segment I regions are removed. The simplified cloak (inset of Fig. S2b) is
constructed with only Segment II regions, with the orientations of their optical axes
the same as those in the original cloak. From the k-surface diagram, if we want to
make the cloak still work for the red ray, the refracted ray represented by S3 should
propagate along the horizontal direction. At the B2(B3) interface, the refraction law
requires the k vectors to have the same value of the component along the tangential
direction. We can inversely get k1 of the incident ray. Further, at the interface of B1,
we need to increase the density of the background medium (i.e. to increase the radius
of the cyan circle) in order to keep S0 along the horizontal direction (Fig. S2b). This is
to say, in order to make the cloak work for the same background medium shown in
Fig. S2a, we can use optical thinner anisotropic crystal which has a smaller elliptical
k-surface as the Segment II medium. The cloak shown in Fig. S2b is constructed with
only one material, without need to consider the Segment I medium, and therefore is
much simpler to implement in the experimental demonstration than the original cloak
(Fig. S2a). A ray tracing analysis on these two cloaks (Fig. S2c and 2d) shows that
both of them can guide the red rays to the other side without any deviation. What we
lost in the simplification is that the original cloak in Fig. S2c works for rays incident
from all the directions, while the simplified cloak in Fig. S2d works for rays only
from six incident directions due to the symmetric hexagonal pattern. The lost
performance of the cloak for other ray directions can be restored if we find highly
anisotropic materials in the future. Nevertheless, considering the current material
availability and the fabrication process, the simplified cloak is very suitable for
experimental demonstration.
References
29. Kong, J. A. Electromagnetic Waves Theory. EMW Publishing, Cambridge, MA,
(2005).