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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

*Jad.Halimeh@physik.lmu.de

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.

References and links

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3.

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C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-

dimensional photonic metamaterials,” Nature Photon. 5, 523-530 (2011).

5.

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss

Limitations for Cloaking of Large Objects,” Phys. Rev. Lett. 104, 253903 (2010).

6.

H. Chena and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep. 2, 255

(2012).

7.

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transformation media,” Opt. Express 14, 9794-9804 (2006).

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medium: a coordinate-free approach,” J. Opt. Soc. Am. A 27, 2558-2562 (2010).

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“Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).

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10. J. C. Halimeh and M. Wegener, “Time-of-flight imaging of invisibility cloaks,” Opt. Express 20, 63-74

(2012).

11. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol. 8

(Butterworth-Heinemann, Oxford, 1984).

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24361-24367 (2010).

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correlation-function-based cloaking-quality assessment,” Opt. Express 19, 6078-6092 (2011).

14. M. Born and E. Wolf, Principles of Optics, 7. Ed. (University Press, Cambridge, 1999).

15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1999).

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Nature Photon. 1, 224-227 (2007).

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three-dimensional carpet cloak,” Phys. Rev. Lett 107, 173901 (2011).

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applications,” Metamaterials 4, 89–97 (2010).

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metamaterials,” Appl. Phys. Lett. 99, 254103 (2011).

1. Introduction

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 [4], 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 [5].

Recent experiments [6] 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. [7] and [8]. We investigate the paradigmatic free-space

cylindrical invisibility cloak [9] 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 [10] 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

[11-13].

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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 [13]. 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

becomes

!!

0

0

! =

0

!!

0

0

0

!!

. (1)

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. [13] we have derived the

corresponding ray equation of motion for the ordinary-ray velocity or energy velocity !! as

!!!

!!.

!!=

!!! ∇!!!! ∇!!∙ !! !!

!!

. (2)

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. [10], 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. [10]. 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

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! !!!!= !!

!! !!=

!

!!

!!!. (3)

Inserting Eq. (3) into the extraordinary ray velocity !! (see Eq. (4) in Ref. [10]) gives

!!×!!

!!

!!

with vacuum permittivity !! (≠ !!), and the electromagnetic energy density !!. Using the

mathematical identity [10]

!!==

!!!×!!

!!

=

!!

!!!!

!

! !! !! × ! !! !!

!!

, (4)

! ! × ! ! = ! !!!!×! , (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 [10] and working out the Euler-Lagrange equations

[10] leads us to the final extraordinary ray equation of motion (see Eq. (11) in Ref. [10]) in

compact form analogous to Newton’s second law [10]

!!!

! (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. [10], but the auxiliary matrix ! for

non-magnetic uniaxial dielectrics in Eq. (7) is generally different from Eq. (8) in Ref. [10].

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

previous section.

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 [15]

! !! =

!!!!

!!

!!"#!!!!!!

!!"#!(!!)

, (9)

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

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!!= 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 [14] 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 [16]

!!=

! ∙ (!! × !!)

! ∙ (!! × !!) 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 [9]

with inner radius ! and outer radius ! are given by [9]

!!= !!=

!!!

! ; !!= !!=

!

!!! ; !!= !!=

!

!!!

!!!!

! . (13)