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2 The present review covers the scattering of plane electromagnetic waves on spherical objects The results shown here might be extended to any arbitrary e.m. wave, expressed as a superposition of time-harmonic waves The applications covered in detail are: scattering resonances on an optically levitated single sphere designing and evaluating metamaterials for invisibility cloaking

Elastic scattering of light may be modeled as an interaction between an electromagnetic wave and an ensemble of electric dipoles On the basis of electromagnetic theory Mie obtained a rigorous solution for the diffraction of a plane monochromatic electromagnetic wave by a homogeneous sphere of any diameter and of any composition situated in a homogeneous medium. 3

Talking about e.m. fields implies using Maxwell's equations language In these formulas    is the electric permittivity in vacuo;  is the electric susceptibility, the amplitude of the response to a unit field;   is the oscillation frequency of the harmonic electromagnetic field;  is the frequency-dependent complex conductivity (for long wavelengths  is, to a good approximation, real);  the extinction coefficient 4

No free charges; harmonic wave The crucial idea in solving the scattering problem is to find a basis of appropriate wave functions. It should match both the incoming field and the symmetry of the boundary. 5 Homogeneous and isotropic medium Harmonic wave equation Complex harmonic field:

Building an orthogonal vector wave basis for the scattering on a sphere 6 P(R,  k For a plane wave whose propagation vector is

7 Mie’s spherical vector wave basis The normalized Bessel functions describe the radial behavior.

Each field is decomposed in linear independent TM and a TE components TE TM 8

Sharp resonances for the reflection coefficients when the incident frequency (real) is near to the scattering coefficients poles ! (complex frequencies= natural frequencies) a_n coefficients describe TE modes b_n coefficients describe the TM modes TE modes and TM modes are decoupled 9 Notations for the boundary conditions

k i : incident wave-vector; it is chosen parallel with z-axis as usually in the experimental settings k s : scattered field wave-vector; the detector is placed in this direction k i and k s determine the scattering plane, which is highlighted  :scattering angle, in the scattering plane; in the the perpendicular plane  : azimuth angle The fields have an in-plane transverse component E p and a component E s perpendicular to the scattering plane. Light scattering on a single sphere and standard decomposition in two perpendicular fields for measurements purposes 10

Ashkin and Dziedzic proving Mie’s theory on a single levitated sphere ~ 2 ~ 2 z(x,x)y z(x,z)x 11 There are two exceptional directions for which the secondary field is linearly polarized: when  =0,    ; if the detector is placed along the axis of incident plane wave polarization in the geometry z(x,z)x, only the component E  is detected; when  2,    ; if the detector is placed perpendicular on the axis of incident plane wave polarization in the geometry z(x,x)y, only the component E  is detected. EOM: electro optic modulator controlling laser input power S2, S3, S4 :screens for near-field view A2, A3, A4: apertures for increased resolution in far-field AR: anti-reflection coating for avoiding interferences on D1 P: tilted plate to avoid backscattering D1,D2,D3,D4: detectors Mic1: projects particle image Mic2: collects backscattered light Mic3: collects the “p” scattered light Mic4: collects the “s” scattered light Focusing lens: f= 60mm Levitated silicone oil drop radius a=11.4  m refractive index N=1.4 P~15mW

Using the wavelength and size dependence of light scattering, Ashkin and Dziedzic demonstrated the resonances in Mie scattering from a single optically trapped liquid droplet 2 2 ~ ~ The theory predicts that sharp resonances for the reflection coefficients appear when the incident frequency (real) is near to the scattering coefficients poles (complex frequencies). All these resonances were found experimentally 12 Comparison of (A) and (B) shows that each characteristic sequence of resonances observed in the levitating power is also detected in the far-field backscattered. The data from (C) and (D) make clear two distinct polarization classes, perpendicular and parallel resonances. artifacts – Gaussian beam

13 Invisible objects: cloaking It is possible to conceal a region of space if surround it with an material displaying anisotropic electric and magnetic properties in such a way that the rays are curved circumventing the inner space to be hidden Instead of considering the permittivity and permeability of materials made out of atoms or molecules, designing metamaterials implies building nano-scale, subwavelength, artificial heterostructures which will give at macroscopic scale the desired refractive index behavior.

14 Periodic heterogeneities give novel electric and magnetic properties The first cloaking metamaterial is made out of radially arranged cells. A plot of the material parameters that are implemented is superimposed on the picture.  r, with a red line, is multiplied with a factor of ten for clarity;    1=const is the green line; e z =3.423=const. is the blue line. There are ten concentric rings, each of them containing slightly different unit cells shown in the insert. The apparatus symmetry, with the electric field polarized along the cylinder axis, accepts a reduced set of material parameters; there is a nonzero calculated reflectance.

15 Each unit cell is a split ring resonator whose resonance frequency can be modified conveniently by tailoring the geometrical parameters. The refractive index of the metamaterial changes in the radial direction while being constant in the transverse direction. The geometrical parameters were taken from numerical simulations. The inner cylinder is 1, the outer cylinder is 10. Unit cell for a cylindrical cloak in microwave domain s: split parameter r: radius of the corners

16 Designing and demonstrating electromagnetic cloaking Performances for the first cylindrical cloak in microwave domain. (A)and (B) show simulations for exact, respectively reduced material properties. (C) and (D)are experimental measurements for the bare, respectively for a cloaked copper cylinder.

17 Mie’s theory applied in cloaking design Heterogeneous medium properties The by-the-date analytical demonstrations for cloaking were mostly in the geometrical optics or in the electrostatic/magnetostatic limit. They included approximations for Maxwell's theory in demonstrating zero scattering cross section for any wavelength condition. Recently, a new theoretical approach using a full wave Mie scattering model for a sphere-like cloak came from a MIT-based team.

18 Each field is decomposed in linear independent TM and a TE components TE electric field TM magnetic field

19 Adjusting standard Mie’s equations for an inhomogeneous and anisotropic medium radial equation homogeneous sphereanisotropic cloak wave vector for a homogeneous and isotropic medium inside the sphere wave vector for the anisotropic cloak and

20 transverse equation Adjusting standard Mie’s equations for an inhomogeneous and anisotropic medium are the associated Legendre polynomials

21 Perfect material behavior  t =  0 [R 2 /(R 2 −R 1 )]  r =  t [(r−R 1 )/2r 2 ] μ t = μ 0 [R 2 /R 2 −R 1 ] μ r = μ t [(r−R 1 )/2r 2 ] R 1 = 0.50 and R 2 = anisotropic cloak Ex field distribution and Poynting vectors due to an Ex polarized plane wave incidence onto an ideal cloak

22 Imperfect material behavior simulated by changing the numerical value for a single parameter a) ordonate: abscise: normalized to For each plot one parameter is kept constant. Case III: the refractive index Case II: the impedance Case I: When Q scatt =0 the cloak is perfect. Case III:  t,norm =2;  t,norm =1/2

23 For solving the problem of scattering of light on a spherical object, Mie’s idea was to decompose the incident, the transmitted, and the reflected wave in linear independent TM and TE components The theory predicts sharp resonances for the reflection coefficients that occur when the incident frequency (real) is near to the scattering coefficients poles (complex frequencies). All these resonances were found experimentally For solving the problem of invisibility, Pendry’s idea was to theoretically model anisotropic materials behaving as cloaks A MIT team of theoreticians is employing both Mie’s and Pendry’s ideas for exact analytical modeling a cloak and better simulating imperfect material behavior