JW4A.49.pdf
Frontiers in Optics 2017 © OSA 2017
A Full Three-Dimensional Isotropic Carpet Cloak Designed by Transformation Optics Daniely G. Silva1 , Poliane A. Teixeira1 , Lucas H. Gabrielli2 , Mateus A. F. C. Junqueira1,∗ , Danilo H. Spadoti1 2 School
1 Federal University of Itajub´ a, Itajub´a, MG, Brazil of Electrical and Computer Engineering, University of Campinas, Campinas, SP, Brazil ∗
[email protected]
Abstract: This work presents a 3D isotropic carpet cloak designed via transformation optics. A quasi-conformal mapping was obtained using numerical optimization. Simulations demonstrate the invisibility effect for arbitrary directions of incident waves. OCIS codes: (130.3120) Integrated optics devices; (230.3205) Invisibility cloaks; (000.3860) Mathematical methods in physics
1.
Introduction
Transformation Optics (TO) is a technique which enables designing of complex optical devices. The control over the propagation of electromagnetic waves depends on the adopted coordinate transformation, under which Maxwell’s equations are invariant. Despite its advantages, TO often results in unconventional material requirements, such as negative refractive index, or inhomogeneous medium with anisotropic permeability and permittivity. Quasi-conformal mapping is a solution to avoid some of these hard requirements, allowing one to treat the materials as if they were isotropic [1]. Most techniques used to obtain such mappings are exclusively two-dimensional (2D). Extensions to threedimensional (3D) geometries are usually obtained by extrusion or revolution process of a 2D refractive index mapping [2]. This is an approximation that leads to a proper propagation control only for waves traveling in the symmetry planes defined by these processes. Therefore, a 3D carpet cloak using such technique will only provide invisibility effects for waves traveling along those planes of symmetry. However, the methods proposed in [3] and [4] confirm that the 3D quasi-conformal mapping can be implemented with TO, resulting in negligible anisotropy and propagation control for any direction of wave vector. In this paper the possibility of designing a full 3D isotropic carpet cloak is demonstrated. The quasi-conformal mapping is obtained using the technique proposed in [4], and the TO is applied in a parallelepiped domain with |x| ≤ w21 , |y| ≤ w22 , and 0 ≤ z ≤ h, which corresponds to the cloak region. The variables w1 , w2 , and h are the cloak width, depth, and height, respectively. The coordinate transformation functions used in the design are: p
q
x0 (x, y, z) = x + b(x, y, z) ∑ ∑
r
∑ Ai jk xi y j zk
i=0 j=0 k=0
p
q
y0 (x, y, z) = y + b(x, y, z) ∑ ∑
r
∑ Bi jk xi y j zk
i=0 j=0 k=0
p q r z πx πy cos cos + b(x, y, z) ∑ ∑ ∑ Ci jk xi y j zk z0 (x, y, z) = z + c 1 − h w1 w2 i=0 j=0 k=0
(1)
The terms before b(x, y, z) represent the initial transformation, whose variable c is the carpet deformation height. The indexes p, q and r are the power series orders, and Ai jk , Bi jk and Ci jk are the optimization parameters. The boundary function b(x, y, z) is chosen to vanish at the boundaries, in order to restrict the parameterization and optimization effects in the cloak, keeping the boundary conditions present in the initial transformation functions: πx πy 2 2 b(x, y, z) = (x + y + z)(z − h) cos cos (2) w1 w2 These conditions are responsible to preserve the cloak functionality described by the map (0, 0, 0) → (0, 0, c), as well as to ensure the continuity of coordinate transformation at the interfaces, resulting in a reflectionless medium in these limits. After applying the quasi-Newton numerical optimization method, the residual anisotropy can be ignored and the transformed region can be considered as isotropic and inhomogeneous. Consequently, any object situated below the deformed reflecting surface of the carpet cloak will become invisible to an external observer.
JW4A.49.pdf
2.
Frontiers in Optics 2017 © OSA 2017
Results
The transformation parameters are c = 0.2 µm, h = 1.5 µm, w1 = w2 = 4 µm, and p = q = r = 3. The background medium has a refractive index n = 1.5, and the lower boundary of device is a Perfect Electric Conductor (PEC). The wavelength used in the simulations was 750 nm, and a graded refractive index profile, varying from 1.45 to 2.24, is presented in Fig. 1a. Therefore, this result ensures the possible fabrication in a silicon-on-insulator (SOI) platform. A reduction of anisotropy is observed with increasing the polynomial series orders, whereas the contrast of refractive indexes is increased, in agreement with [1, 4]. The 3D simulations are performed using two Gaussian beams sources, with different directions of the wave vector, i.e., one in the (y,z) plane and the other in a orthogonal (x,z) plane. They traveling towards the point (x = 0, y = 0, z = 0) at 45° of incidence, and reflecting at the lower boundary of the carpet cloak. The right side of Fig. 1 presents the normalized electric field propagation in perspective and frontal views of each cut planes. As a reference, Fig. 1b shows the reflection from a perfectly flat floor. The presence of the PEC deformation defined by the boundaries of (1) distort the reflection in Fig. 1c, which means invisibility loss. It is restored by the introduction of the optimized isotropic material that constitutes the cloak, indicated by the dashed lines, in Fig. 1d. The preservation of electric field propagation for both Gaussian excitations proves the possibility of achieving invisibility in different directions.
Fig. 1. Simulation results for the optimized 3D isotropic carpet cloak. (a) Refractive index profile with p = q = r = 3. Perspective view, and planes x = 0 and y = 0 for: (b) a perfectly flat floor, (c) deformed floor without cloak, and (d) deformed floor with cloak.
3.
Conclusion
The parameterization and optimization techniques, applied to TO, successfully produced a full 3D isotropic carpet cloak, that creates an invisibility effect independently of the propagation direction of the incident electromagnetic waves. A medium with negligible anisotropy and reflectionless was obtained, enabling to be integrated in a silicon platform. 4.
Acknowledgments
This work was supported by CAPES, CNPq and FAPEMIG. References 1. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Physical Review Letters, vol. 101, no. 20, p. 203901, Nov. 2008. 2. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett., vol. 105, p. 193902, Nov 2010. 3. C. Garca-Meca, R. Ortuo, J. Mart, and A. Martnez, “Full three-dimensional isotropic transformation media,” New Journal of Physics, vol. 16, no. 2, p. 023030, 2014. 4. M. A. F. C. Junqueira, L. H. Gabrielli, F. Beltr´an-Mej´ıa, and D. H. Spadoti, “Three-dimensional quasi-conformal transformation optics through numerical optimization,” Opt. Express, vol. 24, no. 15, p. 16465, Jul 2016.
