Photorealistic ray tracing of free-space
invisibility cloaks made of uniaxial dielectrics
Jad C. Halimeh1,* and Martin Wegener2
1Physics Department and Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München, D-80333 München, Germany
2Institut für Angewandte Physik, DFG-Center for Functional Nanostructures (CFN),
and Institut für Nanotechnologie, Karlsruhe Institute of Technology (KIT),
D-76128 Karlsruhe, Germany
Abstract: The design rules of transformation optics generally lead to
spatially inhomogeneous and anisotropic impedance-matched magneto-
dielectric material distributions for, e.g., free-space invisibility cloaks.
Recently, simplified anisotropic non-magnetic free-space cloaks made of a
locally uniaxial dielectric material (calcite) have been realized
experimentally. In a two-dimensional setting and for in-plane polarized light
propagating in this plane, the cloaking performance can still be perfect for
light rays. However, for general views in three dimensions, various
imperfections are expected. In this paper, we study two different purely
dielectric uniaxial cylindrical free-space cloaks. For one, the optic axis is
along the radial direction, for the other one it is along the azimuthal
direction. The azimuthal uniaxial cloak has not been suggested previously to
the best of our knowledge. We visualize the cloaking performance of both
by calculating photorealistic images rendered by ray tracing. Following and
complementing our previous ray-tracing work, we use an equation of
motion directly derived from Fermat’s principle. The rendered images
generally exhibit significant imperfections. This includes the obvious fact
that cloaking does not work at all for horizontal or for ordinary linear
polarization of light. Moreover, more subtle effects occur such as viewing-
angle-dependent aberrations. However, we still find amazingly good
cloaking performance for the purely dielectric azimuthal uniaxial cloak.
OCIS codes: (080.0080) Geometric optics; (230.3205) Invisibility cloaks; (160.3918)
Metamaterials; (080.2710) Inhomogeneous optical media.
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Transformation optics maps the geometry of a fictitious space onto actual material properties
in the laboratory [1-3]. Invisibility cloaking continues to be a fascinating benchmark example
to test these ideas. Generally, spatially inhomogeneous and anisotropic magneto-dielectric
material distributions result. Equal magnetic and dielectric responses are required at the same
time to have (i) anisotropic light propagation yet no polarization dependence of the optical
response and (ii) no reflections from interfaces via matching of the relative optical impedance,
which is given by the square root of the ratio of the magnetic permeability ! and the electric
permittivity !. Obtaining an effective magnetic response at optical frequencies has become
possible via three-dimensional metamaterials , but is necessarily connected with
resonances. Hence, for passive structures, dispersion and finite losses via causality and the
Kramers-Kronig relations are unavoidable and often unacceptable, especially in the context of
macroscopic cloaking .
Recent experiments  on macroscopic broadband visible-frequency free-space
invisibility cloaks made of standard uniaxial calcite have used purely dielectric anisotropic
materials (calcite), i.e., the magnetic permeability is set to unity everywhere. This means that
the response becomes polarization dependent and not impedance-matched.
It is clear that completely neglecting the magnetic response and using uniaxial instead of
biaxial materials are rather drastic ad hoc approximations. Apart from severely easing the
experimental realization, these approximations are motivated by the fact that the behavior
remains ideal for propagation of light in a two-dimensional plane and for linear polarization of
light lying in that same plane. In this paper, we visualize the aberrations that occur as a result
of these approximations for both polarizations of light and for more general viewing
conditions in three dimensions by ray tracing. Early work on ray tracing in transformation
media has been published in Refs.  and . We investigate the paradigmatic free-space
cylindrical invisibility cloak  with a continuously varying material distribution of a uniaxial
electric permittivity tensor and unity magnetic permeability as an example. This example
enables direct comparison with our previous work  on corresponding ideal impedance-
matched magneto-dielectric cylindrical cloaks. In Sect. 2, we again use ray equations of
motion for the ordinary and the extraordinary rays derived directly from Fermat’s principle
At the interfaces of the structure, double refraction occurs. While the mathematical
treatment may be seen as “straightforward”, we do summarize the general formulas in Sect. 3
in compact form. In the literature, usually only special cases are explicitly discussed [14-17].
Sect. 4 presents rendered images on the basis of Sect. 2 and 3. In Sect. 5 we explain the reason
behind the significant difference in cloaking performance between the two variants of the
uniaxial dielectric cylindrical cloak.
2. Ray equation of motion in non-magnetic locally uniaxial material distributions
In the geometrical optics of birefringent uniaxial dielectric materials, one has to distinguish
between the paths of the ordinary and the extraordinary rays . What polarization of light is
ordinary and what is extraordinary will generally change at the interface between two
different birefringent materials. However, if the change of the anisotropy axis ! is continuous
in space, the ordinary ray stays ordinary and the extraordinary ray stays extraordinary
throughout even though the ! axis changes. Only two different rays emerge from the overall
structure (apart from further reflected contributions). We will restrict ourselves to this
conceptually transparent case. Only at the interfaces of air/vacuum to the continuously
changing medium, caution has to be exerted. The corresponding discussion will be given in
the next section.
Inside of the structure, we can choose the local Cartesian coordinate system such that,
e.g., the local z-axis coincides with the local ! axis. In this basis, the local dielectric tensor
Here we have suppressed the dependence on ! for better readability and we will continue
doing so throughout this section. As usual, the index “o” stands for ordinary and “e” for
extraordinary here and below. In addition, we assume no magnetic response at all, i.e., we
have the magnetic-permeability tensor ! = 1. The ordinary and extraordinary refractive
indices are then given by !!=!! and !!=
Let us start by discussing the propagation of the ordinary ray, for which the polarization
of light, i.e., the orientation of electric-field vector !!, is perpendicular to the local ! axis
everywhere in the cloak. As a result, this ordinary ray merely experiences a simple refractive-
index distribution equivalent to a ray in a locally isotropic yet spatially inhomogeneous
medium (= “graded-index” structure). In Eq. (11) in Ref.  we have derived the
corresponding ray equation of motion for the ordinary-ray velocity or energy velocity !! as
!!! ∇!!!! ∇!!∙ !! !!
It should be emphasized, however, that the ordinary rays do not lead to cloaking for the
cylindrical cloak to be discussed in Sect. 4.
Thus, the orthogonally polarized extraordinary rays are more important under our
conditions. Their ray equation of motion is more involved though. We can closely follow the
spirit of the derivation in Sect. 3 of Ref. , but we need to replace the expression
!!= !!! !!= !!! !! (with vacuum permeability µμ!) there by an appropriate expression to
again arrive at a compact form like Eq. (7) in Ref. . Under the present conditions, the
extraordinary magnetic field vector !! is perpendicular to the local ! axis everywhere in the
cloak. Thus, its direction is invariant under multiplication with ! (and, therefore, also ! !!),
just like the ordinary displacement vector !!, which is also perpendicular to !. With Eq. (1),
this allows us to write
! !!!!= !!
Inserting Eq. (3) into the extraordinary ray velocity !! (see Eq. (4) in Ref. ) gives
with vacuum permittivity !! (≠ !!), and the electromagnetic energy density !!. Using the
mathematical identity 
! !! !! × ! !! !!
! ! × ! ! = ! !!!!×! , (5)
where … denotes the matrix determinant, we can connect the extraordinary ray velocity to
the extraordinary wave vector of light !! via
!!!! ! !!! !! , (6)
with the angular frequency of light ! and the dimensionless auxiliary matrix
! ! ! !!= !!
! ! !!. (7)
In the last step in Eq. (7) we have inserted the determinant ! = !!
all quantities in Eq. (7) are generally still dependent on the coordinate vector !. Inserting Eq.
(6) and Eq. (7) into Fermat’s principle  and working out the Euler-Lagrange equations
 leads us to the final extraordinary ray equation of motion (see Eq. (11) in Ref. ) in
compact form analogous to Newton’s second law 
! (see Eq. (1)). Note that
! !!! ! !!− 2! ! ∙ !! !! . (8)
Obviously, this ray equation of motion is of the same form as that derived for impedance-
matched anisotropic magneto-dielectrics Eq. (11) in Ref. , but the auxiliary matrix ! for
non-magnetic uniaxial dielectrics in Eq. (7) is generally different from Eq. (8) in Ref. .
3. Interface of the structure to air/vacuum
Arbitrarily polarized incident light impinging from the outside of the cloak (i.e., from vacuum
or air) will be double refracted at the cloak’s interface into an ordinary and an extraordinary
ray, which will then propagate inside the cloak. Thus, we first need to determine the
amplitudes and directions of these two rays before being able to apply the results of the
In the following, the general case is considered where the optic axis ! is not necessarily in
the plane of incidence, which is the case in the azimuthal uniaxial cloak discussed in the next
section. Two planes are defined: the ordinary main section, formed by the optic axis and the
ordinary refracted wave vector, and the extraordinary main section, formed by the optic axis
and the extraordinary refracted wave vector. The ordinary wave has its polarization vector
orthogonal to the ordinary main section, while the extraordinary wave has its polarization
vector in the extraordinary main section. The ordinary wave thus sees a refractive index equal
to !! whereas the refractive index ! that the extraordinary wave sees is given by 
! !! =
where !! is the angle the extraordinary polarization vector includes with the unit optic axis !.
If !! is the angle of refraction of the extraordinary wave, one determines that
!!= arcsin! ∙ ! sin(!!) − ! ∙ ! cos(!!) , (10)
where ! is the unit normal vector and ! is the unit tangential vector in the plane of incidence
(one can see that in the case of the radial uniaxial dielectric cloak, where the optic axis is
parallel to the unit normal, we have, e.g., !!= !/2 − !!). As such, ! = !(!!) can now be
expressed as a function of !!, which enters into the equation of phase matching
!!sin !! = !!sin !! = ! !! sin (!!), (11)
where !!=1 is the refractive index of air/vacuum, !! is the angle of incidence, and !! is the
angle of ordinary refraction. From Eq. (11), it is straightforward to calculate !!. To determine
!!, one must solve the implicit Eq. (11) from which two solutions emerge, with only one of
them acceptable being in [0,!/2]. As such, the directions of the refracted waves are
determined. The amplitudes of the wave vectors are proportional to their respective refractive
indices due to the condition of phase matching, and therefore the ordinary wave vector will
carry a magnitude !!= !!!/!! (with the vacuum speed of light !!) and that of the
extraordinary wave a magnitude of !!= ! !!!/!!.
Note that the above procedure allows determining the directions of all fields since the
directions of the wave vectors are now known and so are the geometric conditions for their
fields. For example, taking into account the extraordinary wave, it is known that its
polarization !! must lie in the extraordinary main section. Moreover, a consequence of
Maxwell’s equations is that !! is perpendicular to !!, which exactly determines its direction
within that plane. Moreover, another consequence of Maxwell's equations is that !!⊥ !!,
and, as the medium is non-magnetic, this means that !!⊥ !!. Since !! is also perpendicular
to !! due to Maxwell’s equations, !! is then perpendicular to the extraordinary main section,
thus exactly determining its direction. With the directions of all fields determined, the
boundary conditions of continuity  then allow us to determine their exact magnitudes. The
ray-velocity vector can then be determined as per Eq. (4) or Eq. (6). The intensity
transmission coefficients !! and !! for the ordinary and extraordinary refracted rays,
respectively, are given by 
! ∙ (!! × !!)
! ∙ (!! × !!) and !!=
! ∙ (!! × !!)
! ∙ (!! × !!) , (12)
where the index “i” stands for the incident quantity.
Although mathematically similar, it is worth briefly mentioning that in the case of an
interface from a uniaxial dielectric crystal to air/vacuum, there will in general be double
reflections within the crystal, into an ordinary reflected wave and an extraordinary reflected
wave, irrespective of whether the incident wave is itself ordinary or extraordinary. The
directions and amplitudes of all relevant fields are determined just as in the case of the
air/vacuum-cloak interface, except here there are two reflections (in cloak) and one refraction
(into air), instead of one reflection (in air) and two refractions (in cloak) at the interface.
Moreover, here it is the reflected wave vectors (ordinary and extraordinary) that, along with
the optic axis, define the main sections.
The intensity reflection coefficients are unity minus the transmission coefficient.
4. Numerical results
In cylindrical coordinates, the ideal magneto-dielectric parameters for the cylindrical cloak 
with inner radius ! and outer radius ! are given by 
! ; !!= !!=
!!! ; !!= !!=
! . (13)