Sub-wavelength image manipulating through
compensated anisotropic metamaterial prisms
Junming Zhao, Yijun Feng*, Bo Zhu, Tian Jiang
Department of Electronic Science and Engineering, Nanjing University, Nanjing, 210093, CHINA
*Corresponding author: firstname.lastname@example.org
Abstract: Based on the concept of sub-wavelength imaging through
compensated bilayer of anisotropic metamaterials (AMMs), which is an
expansion of the perfect lens configuration, we propose two dimensional
prism pair structures of compensated AMMs that are capable of
manipulating two dimensional sub-wavelength images. We demonstrate that
through properly designed symmetric and asymmetric compensated prism
pair structures planar image rotation with arbitrary angle, lateral image shift,
as well as image magnification could be achieved with sub-wavelength
resolution. Both theoretical analysis and full wave electromagnetic
simulations have been employed to verify the properties of the proposed
prism structures. Utilizing the proposed AMM prisms, flat optical image of
objects with sub-wavelength features can be projected and magnified to
wavelength scale allowing for further optical processing of the image by
©2008 Optical Society of America
OCIS codes: (120.4570) Optical design of instruments; (160.1190) Anisotropic optical
materials; (160.3918) Metamaterials; (230.5480) Prisms; (260.2110) Electromagnetic optics.
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One of the major obstacles in optics and photonics is the diffractive nature of light that has
limited us to manipulate images at scales less than the wavelength of light [1,2]. Recently, the
emerging field of metamaterials has provided ways to design artificial materials with unusual
optical properties and such diffraction limit for imaging has been overcome through the
perfect lens structure proposed by J. Pendry . A planar slab of lossless left-handed
metamaterial (LHM) with simultaneously negative permittivity and permeability can be made
as a perfect lens, where both propagating and evanescent waves emitted from a given light
source could be recovered completely at the image points inside and outside the slab.
Although for practical realization, its sensitivity to material loss and other factors can limit the
sub-wavelength resolution [4-7], the concept was validated experimentally by using a thin
slab of structured LHM  at microwave range, or silver [9-10] and SiC  at optical range.
In addition to isotropic LHM, anisotropic metamaterials (AMMs) having permittivity and
permeability tensors with parts of the elements being negative have drawn a lot of attentions
due to their extraordinary properties, such as negative refraction [12,13], partial focusing ,
and resonance-cone focusing [15,16]. The AMMs have been identified into four types based
on their wave propagation properties, which are called cutoff, always-cutoff, never-cutoff and
anti-cutoff media . Compensated bilayer structure of such AMMs has been proposed as an
expanded perfect lens configuration, which could transfer the field distribution from one side
of the bilayer to the other with sub-wavelength resolution not restricted by the diffraction limit
[17,18]. Both theoretical analysis  and experimental verification  show that although
such compensated bilayer lens provides no free space working distance, it could produce
image with an enhanced resolution that exhibit a decreased sensitivity to losses and to
deviations in material parameters relative to the LHM perfect lens configuration.
However, it has been pointed out that the perfect lens structure can only work for the near-
field , which makes the image difficult to be processed or brought to focus by
conventional optics. To solve this problem, a new sub-diffraction-limit imaging method called
‘hyper-lens imaging’ has been proposed [22-24] and experimentally verified [25-26], which
can resolve and magnify sub-wavelength details utilizing the unusual optical phenomenon of
strongly anisotropic metamaterials. The hyper-lens can project the magnified image into the
far field - where it can be further manipulated by the conventional (diffraction-limited) optics.
It should be mentioned that due to the cylindrical structure, such hyper-lens can only transfer
image between the inner and the outer circular cylinder boundaries, which limits its optical
applications. Further improvements have been reported by using the concept of coordinate
transformation to design planar magnifying perfect-lens  or hyper-lens , but these
theoretical proposals require complicated metamaterial with spatial varying anisotropic
material parameters that are difficult to realize.
In this paper, we expand the concept of sub-wavelength image through compensated
bilayer of AMMs. We propose two dimensional (2D) prism pair configurations of
compensated AMM bilayer that are capable of manipulating sub-wavelength images. We
demonstrate that image rotation with arbitrary angle, lateral image shift and image
magnification could be achieved with sub-wavelength resolution through properly designed
prism structures. Both theoretical analysis and full-wave electromagnetic (EM) simulations
have been employed to verify the properties of the proposed prism structures. Utilizing the
compensated AMM prisms, planar optical image of sub-wavelength objects can be magnified
to wavelength scale allowing for further optical processing of the image by conventional
2. Sub-wavelength imaging using 2D compensated bilayer of AMMs
Pendry’s perfect lens can be viewed as a bilayer of vacuum and LHM, in which the LHM
exactly compensates for the propagation effects associated with an equal length of vacuum.
This concept of compensated bilayer has been extended to different types of AMM pairs
[12,18], and 2D perfect lens structures could be accomplished by using AMMs for which
some principal components of the permittivity and permeability tensors have negative value.
For example, compensated bilayer of anti-cutoff (ACM) or never-cutoff medium (NCM)
[12,19] could act as a perfect lens that transfers the electromagnetic field at the front surface
of the bilayer to the back surface completely in both magnitude and phase to accomplish sub-
To understand the mechanism, we theoretically analyze the EM propagation when a 2D
point source is placed outside a bilayer of AMM as shown in Fig 1. Assuming Region 0 and
Region III are free space, while Region I and Region II are filled by AMMs with material
tensor denoted as
zz . (j = 1, 2). (1)
j jx jyjzj jx jyjz
ε ε εεεμ μ μμμ=++=++
To study the sub-wavelength imaging phenomenon, we assume the source is in front of the
front interface of the bilayer with d1 → 0. Following the classical electromagnetic theory, the
fields radiated by the point source can be expressed as closed forms of Sommerfeld-type
Fig.1. 2D point source propagating through a compensated AMM bilayer.
Region I Region II
The transmission coefficient T of the electric field at the back interface of the bilayer can
be calculated as 
= 8[(1+ )(1+ )(1+ )ep
+(1)(1 )(1+ )e+(1 )(1+ )(1)e]
i k L +k Li k L k L
i k L k L i k L +k L
where p, q, r are the relative effective impedances defined as
pkkqkkr k k
and k0x , k1x or k2x are the x component of the wave vector in free space, the first or the second
layer of the bilayer, respectively, which satisfy different dispersion relations
yyjx jx jy jzjx
μμ ε μ+=+=
, (j = 0, 1) (4)
To obtain a perfect image at the back side of the bilayer, a unit transmission coefficient, T = 1,
is required for all values of the transverse wave vector ky. From Eq. (2), when q = 1, and k1xL1
+ k2xL2 = 0 (k1x and k2x have opposite sign, representing propagating mode), or q = −1, and
k1xL1 − k2xL2 = 0 (k1x and k2x have the same sign, representing evanescent mode), T becomes
unity resulting in a compensated bilayer .
Obviously, in addition of building compensated bilayer with vacuum and anti-vacuum (ε =
−ε0 and μ = −μ0), which yields the configuration of Pendry’s perfect lens, we could also build
compensated bilayer by combining positive and negative refracting layers of NCM or ACM.
The electromagnetic field at the front surface of the bilayer could be restored completely in
both magnitude and phase at the back surface accomplishing near-field focusing. But we
should emphasis the difference between isotropic Pendry’s perfect lens and the AMM bilayer
perfect lens. For example, in the case of NCM bilayer lens, both the propagating and the
evanescent components of the source are converted into propagating modes in the NCM
bilayer, and then back to propagating and evanescent components on the back surface,
building an image with resolution beyond the diffraction limit.
The above analysis also allows us to construct compensated bilayer with unequal layer
thickness. Assuming the thickness ratio η = L2/L1, compensated bilayer of NCM or ACM for
both TE and TM waves requires
εμεμ α β η α
γ α γ α
where α1α2 < 0, and α1, α2, β1, γ are four arbitrary parameters that determine the material
tensor elements of the compensated bilayer. In Eq. (5), γ > 0 stands for an ACM compensated
bilayer, while γ < 0 stands for a NCM compensated bilayer.
We rigorously calculate the EM wave propagation of point sources through AMM
compensated bilayer and two examples are demonstrated in Fig. 2. The first example is an
equal thickness NCM bilayer with material parameters chosen as μ1x = − μ1y = −1, μ2x = − μ2y
= 1, and ε1z = −ε2z = 1, and the second example is an unequal thickness NCM bilayer with
parameters chosen as μ1x = − μ1y = 1, μ2x = − 3, μ2y = 1/3, ε1z = −1, and ε2z = 1/3. In both case
small electric and magnetic loss tangent of 10-3 are included to see their influences on the
image quality. Excited with the two point sources that are separated about half wavelength at
the front surface, two clearly focused images are observed at the back surface of the
compensated bilayer as shown in Fig 2 (a) and (c). Unlike the case of an isotropic LHM
planar lens, both the propagating and the evanescent components of the source incident into
the NCM bilayer are converted into propagating modes, and then back to propagating and
evanescent components on the back surface, building an image with resolution beyond the
diffraction limit as shown in Fig. 2 (b) and (d). A standing wave mode is established inside the
bilayer instead of a coupled surface plasmon mode in the case of LHM planar lens.
3. Sub-wavelength image manipulating through compensated AMM prism pairs
3.1 Sub-wavelength imaging between unparallel planes
The Pendry’s perfect lens configuration is restricted to parallel source and image planes. Now
we expand the compensated bilayer structure to a compensated prism pair (CPP) structure
which can manipulate 2D images with sub-wavelength resolution between unparallel planes.
We first consider a symmetry CPP (S-CPP) configuration as shown in Fig. 3 (a). Two
prisms with a same apex angle of θ (Prism1 and Prism2) are symmetrically putting together
forming a prism pair. The two prisms are composed of AMMs with one of their optical axes
(the y axis) aligned with the symmetry axis OO′ and the material permittivity and permeability
tensors satisfy the requirement of forming an equal thickness compensated AMM bilayer (i.e.
Eq. (5) with η = 1). Due to the compensation nature of the two prisms, 2D sub-wavelength
objects at the left surface OA will be perfectly imaged at the right surface OB of the S-CPP
structure. For example, point sources at S1 and S2 of the source plane will be restored at I1 and
I2 of the image plane, respectively, with S1I1 and S2I2 perpendicular to the interface OO′ of the
structure. Such an S-CPP structure acts as an optical image component that makes perfect
image between unparallel source and image planes with 2θ rotation in the limit of sub-
To verify the performance, we carry out full wave EM simulation based on finite element
method of the proposed S-CPP structure. Fig. 3 (b) shows the calculated electric field
Fig. 2. Electric field mapping for two point sources located at the front interface of a
NCM compensated bilayer with a loss tangent of 10-3 ((a), and (c)) and the comparison
of the beamwidth of electric field distribution at the front and back interfaces ((b), and
(d)) for an equal thickness bilayer ((a), and (b) with L2 = L1) and an unequal thickness
bilayer ((c) and (d) with L2 = 3L1), respectively. The white dashed lines indicate the
boundaries of the bilayer.
distribution of two point sources separated about 0.03 wavelength at the left surface of a S-
CPP structure with θ = 30°. The material tensor elements are chosen as μ1x = − μ2x = −3, μ1y =
− μ2y = 1/3, and ε1z = − ε2z = 1/3, and a loss tangent of about 0.01 has been included for each
tensor element. It is clearly demonstrated that at the right surface well resolved images of the
two point sources are obtained, which confirms the ability of the S-CPP structure for imaging
sub-wavelength objects to an unparallel plane. It is worth noting that the sub-wavelength
imaging by S-CPP is not sensitive to small material losses, similar to that of the case of a
compensated bilayer lens [19, 20]. Moreover, the boundary effect on the image quality is
almost unobservable for prisms with finite sizes since most of the EM power is restricted to
the rhombus area between the source and the image as indicated in Fig. 3 (b).
The proposed S-CPP structures can be used as optical components to build more
complicated imaging system. For example, it can be cascaded to make different image
rotation with arbitrary angles, or to produce sub-wavelength image with a lateral translation.
Fig. 4 shows that a shifted image with sub-wavelength resolution can be obtained with a
lateral translation to the original object by a combination of four identical S-CPPs with apex
angle of 2θ = 45°. Two point sources separated by 0.04 wavelength have been well imaged at
the other side of the S-CPP system with a 0.15 wavelength lateral shift.
3.2 Magnified imaging with sub-wavelength resolution
Next we consider more general case of an asymmetry CPP (AS-CPP) configuration as shown
in Fig. 5. Two prisms with apex angles of α (for Prism1) and β (for Prism2) are putting
together forming an asymmetry prism pair. The two prisms are composed of AMMs with one
of their optical axes (the y axis) aligned with the common axis OO′ and the material
permittivity and permeability tensors of the AMM satisfy the requirement of forming an
unequal thickness compensated AMM bilayer (i.e. Eq. (5) with η ≠ 1). Similar to the S-CPP
structures, due to the compensation nature of the two prisms, sub-wavelength objects at the
left surface OA will be perfectly imaged at the right surface OB of the AS-CPP structure. For
example, point sources at S1 and S2 of the source plane will be restored at I1 and I2 of the
image plane, respectively, with S1I1 and S2I2 perpendicular to the interface OO′ of the
structure. The interesting feature of this AS-CPP structure is that the image size is unequal to
that of the object with a magnification determined by
Fig. 3. (a) Schemetic of a symmetry compensated prism pair configuration. (b) Electric
field distribution of two point sources imaged with the S-CPP structure with a loss
tangent of 0.01.
and the two apex angles α and β are restricted by the compensation requirement of the two
AMMs for the prisms, that is the thickness ration η in Eq. (5), which satisfies
1 12 2
Thus, from Eq. (5) – (7) we are able to design certain AS-CPP structure that is possible of
producing magnified image of sub-wavelength objects with in principle arbitrary
As an example, we design an AS-CPP with a magnification of τ = 3. For convenient we
can further restrict α+β = π/2, then we yield η = τ
2 = 9. The two apex angles of the prisms can
Fig. 5. Schematic of an asymmetry compensated prism pair configuration.
Fig. 4. Electric field distribution of two point sources imaged with four identical S-CPP
structures cascaded together with a loss tangent of 0.01. Such configuration could
produce lateral image translation with sub-wavelenth resolution. The white lines indicate
the boundaries of the S-CPPs.
be determined by Eq. (6) or (7), and the material parameters can be designed through Eq. (5).
Actually we can see that there is plenty of freedom for design such an AS-CPP. Here we
choose ε1x = μ1x = −3, ε1y = μ1y = 1/3, ε1z = μ1z = 1/3, ε2x = μ2x = 27, ε2y = μ2y = -1/27, ε2z = μ2z =
-1/27, and a loss tangent of about 0.01 has been included for each tensor element. To verify
the performance, we calculated the electric field distribution (Fig. 6 (a)) for three 2D point
sources imaged with such an AS-CPP which has been cut into a cuboid shape. The material
optical axes of the two prisms are parallel or perpendicular to the interface. The three point
sources placed at the left boundary with unequal separations have been projected to the
bottom boundary achieving a magnified image with sub-wavelength features of the objects.
The line scans at both the source and image planes have also been compared in Fig. 6 (b) and
(c), which clearly indicated a linear magnification of 3. The slightly intensity dimming and
peak broadening from left to right of the image plane is due to the increasing loss as a result of
the extending of the image-object distance, but the sub-wavelength features of the object have
been well resolved.
By cascading the AS-CPPs, we can make sub-wavelength image with larger
magnification. Fig. 7 (a) shows the imaging of three point sources with two AS-CPPs
cascaded together. The materials of the two AS-CPPs are similar to that used in the previous
example, which lead to a total magnification of 9. The simulated electric field distribution is
shown in Fig. 7 (a), and line scans at the source and image planes are illustrated in Fig. 7 (b)
and (c), respectively, which indicate that the three point sources with separations within 0.1
wavelength have been well resolved and magnified to wavelength scale. The cascaded system
also partially compensated the nonuniform image intensity in single AS-CPP
Fig. 6. (a) Electric field distribution for three point sources imaged with an AS-CPP with
a designed magnification of 3. Loss tangent of 0.01 is included for each material
parameter. Line scans at the source (b), and image (c) planes of the electric field which
have been normalized to that of the source value.
resulting in more homogenous image intensity as shown in Fig. 7 (c). Using the AS-CPP
structure, planar objects with deep sub-wavelength features can be projected and magnified to
wavelength scale planar image. Such magnified image can be further processed by
conventional optics, and the both flat object and image planes are more convenient for
imaging and lithography applications.
The material requirements for building either A-CPP or AS-CPP are simply anisotropic
with partial negative permittivity or permeability components and do not need any spatial
variation. Thus they are more achievable compared to recent proposals of designing planar
magnifying perfect-lens  or hyper-lens  based on the concept of coordinate
In this paper, we expand the concept of sub-wavelength imaging through compensated bilayer
of AMMs. We propose 2D prism pair configurations of compensated AMM that are capable
of manipulating sub-wavelength images. We demonstrate that planar image rotation with
arbitrary angle, lateral image shift, as well as magnified image could be achieved with sub-
wavelength resolution through properly designed compensated prism structures. The
theoretical analysis and design procedure of these image processing components have been
given and their performances have been confirmed by FEM based full wave EM simulation.
Utilizing the proposed AMM prisms, planar optical image of objects with sub-wavelength
features can be magnified to wavelength scale allowing for further optical processing of the
image by conventional optics. With the rapid development of the metamaterial design and
fabrication techniques in optical range, we believe the proposed sub-wavelength image
Fig. 7. (a) Electric field distribution for three point sources imaged with two cascaded
AS-CPPs with a total designed magnification of 9. Loss tangent of 0.01 is included for
each material parameter. Line scans at the source (b), and image (c) planes of the
electric field which have been normalized to that of the source value.
manipulations could be applied to optical imaging and lithography systems with sub- Download full-text
This work is supported by the National Basic Research Program of China (2004CB719800),
and the National Nature Science Foundation (No. 60671002).