^{1,a)}, Nathan Landy

^{1}and David R. Smith

^{1}

### Abstract

We propose a generalization of the two-dimensional eikonal-limit cloak derived from a conformal transformation to three dimensions. The proposed cloak is a spherical shell composed of only isotropic media; it operates in the transmission mode and requires no mirror or ground plane. Unlike the well-known omnidirectional spherical cloaks, it may reduce visibility of an arbitrary object only for a very limited range of observation angles. In the short-wavelength limit, this cloaking structure restores not only the trajectories of incident rays, but also their phase, which is a necessary ingredient to complete invisibility. Both scalar-wave (acoustic) and transverse vector-wave (electromagnetic) versions are presented.

This work was supported by the U.S. Navy through a subcontract with SensorMetrix (Contract No. N68335-11-C-0011), and partially by the U.S. Army Research Office through a Multidisciplinary University Research Initiative (Grant No. W911NF-09-1-0539). The authors are grateful to Daniel Smith (COMSOL Inc.) for useful discussions relating to numerical ray-tracing algorithms, and to Nathan Kundtz (Intellectual Ventures) for his comments on the utility of conformal maps in cloaking devices.

I. INTRODUCTION

II. ANALYSIS OF THE TWO-DIMENSIONAL CONFORMAL CLOAK

III. THREE-DIMENSIONAL ISOTROPIC-MEDIUM UNIDIRECTIONAL ACOUSTIC CLOAK

IV. THREE-DIMENSIONAL POLARIZATION-INSENSITIVE ELECTROMAGNETIC CLOAK

V. CONCLUSIONS

### Key Topics

- Refractive index
- 24.0
- Anisotropy
- 10.0
- Acoustic waves
- 8.0
- Boundary value problems
- 8.0
- Metamaterials
- 8.0

##### F41H3/00

## Figures

(Color online) Two-dimensional conformal cloak proposed in Refs. 23 and 24: (a) Refractive index profile, with the boundary between superluminal (*n* < 1) and sub-luminal quadrants shown by solid curves (except the solid circle indicating the cloak boundary at *r* = *a*); the shading inside the circle (*r* = *a*) is cutoff at refractive index values *n* = 0 and *n* = 2, respectively; (b) refractive index as a function of the polar angle on the two sides of the discontinuity at *r* = *a*; (c) ray-tracing simulation showing trajectories and phase of the rays in the eikonal limit, assuming a plane wave incident along the *x* direction; (d) same as (c) with a plane wave propagating in the *y* direction.

(Color online) Two-dimensional conformal cloak proposed in Refs. 23 and 24: (a) Refractive index profile, with the boundary between superluminal (*n* < 1) and sub-luminal quadrants shown by solid curves (except the solid circle indicating the cloak boundary at *r* = *a*); the shading inside the circle (*r* = *a*) is cutoff at refractive index values *n* = 0 and *n* = 2, respectively; (b) refractive index as a function of the polar angle on the two sides of the discontinuity at *r* = *a*; (c) ray-tracing simulation showing trajectories and phase of the rays in the eikonal limit, assuming a plane wave incident along the *x* direction; (d) same as (c) with a plane wave propagating in the *y* direction.

(Color online) Two-dimensional conformal cloak proposed by Leonhardt *et al.* (Refs. 23 and 24). Full-wave simulations of the structure shown in Fig. 1. Angle of incidence: (a) 0, (b) *π*/8, (c) *π*/4, (d) *π*/2. TE polarization (*E* field out of plane) is assumed. Refractive index distribution is implemented using *ε _{z} * =

*n*

^{2}, with

*n*given by Eqs. (2) and (3) and in-plane permeability

*μ*= 1. Free-space wavelength

*λ*

_{0}=

*a*/5.

(Color online) Two-dimensional conformal cloak proposed by Leonhardt *et al.* (Refs. 23 and 24). Full-wave simulations of the structure shown in Fig. 1. Angle of incidence: (a) 0, (b) *π*/8, (c) *π*/4, (d) *π*/2. TE polarization (*E* field out of plane) is assumed. Refractive index distribution is implemented using *ε _{z} * =

*n*

^{2}, with

*n*given by Eqs. (2) and (3) and in-plane permeability

*μ*= 1. Free-space wavelength

*λ*

_{0}=

*a*/5.

(Color online) Three-dimensional unidirectional acoustic cloak consisting of isotropic medium: (a) Pressure distribution on the cross-section of the cloak. Black lines show the streamlines of acoustic flux; acoustic wavelength at *r* *a* is *λ* _{0} = *a*/2; (b) three-dimensional picture of the cloak with a finite exterior radius *R _{cut} * = 4

*a*; wavelength in ambient medium (at

*r*>

*R*)

_{cut}*λ*

_{0}=

*a*/2; the index transition is smoothed on a spatial scale Δ

*R*=

*a*/5. Acoustic wave impedance is assumed constant in both simulations; the speed of sound varies according to

*c*=

_{s}*c*

_{0}/

*n,*where

*n*is prescribed by the revolution of profile (2).

(Color online) Three-dimensional unidirectional acoustic cloak consisting of isotropic medium: (a) Pressure distribution on the cross-section of the cloak. Black lines show the streamlines of acoustic flux; acoustic wavelength at *r* *a* is *λ* _{0} = *a*/2; (b) three-dimensional picture of the cloak with a finite exterior radius *R _{cut} * = 4

*a*; wavelength in ambient medium (at

*r*>

*R*)

_{cut}*λ*

_{0}=

*a*/2; the index transition is smoothed on a spatial scale Δ

*R*=

*a*/5. Acoustic wave impedance is assumed constant in both simulations; the speed of sound varies according to

*c*=

_{s}*c*

_{0}/

*n,*where

*n*is prescribed by the revolution of profile (2).

(Color online) Three-dimensional polarization-insensitive electromagnetic cloak made of isotropic non-magnetic medium: (a) Three-dimensional full-wave simulation with free-space wave-length *λ* _{0} = 0.8*a* showing electric field component transverse to the propagation direction; simulation domain (excluding the PML) radius *R* = 4*a;* (b) 2.5-dimensional model with *λ* _{0} = *a*/2 and cloak exterior radius *R _{cut} * = 6

*a;*transition width Δ

*R*= 0.3

*a.*

(Color online) Three-dimensional polarization-insensitive electromagnetic cloak made of isotropic non-magnetic medium: (a) Three-dimensional full-wave simulation with free-space wave-length *λ* _{0} = 0.8*a* showing electric field component transverse to the propagation direction; simulation domain (excluding the PML) radius *R* = 4*a;* (b) 2.5-dimensional model with *λ* _{0} = *a*/2 and cloak exterior radius *R _{cut} * = 6

*a;*transition width Δ

*R*= 0.3

*a.*

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