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SCATTERING OF LIGHT ON SPHERES

METAMATERIAL CLOAK

Emanuela Ene

Oklahoma State University

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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

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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.

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Talking about e.m. fields implies usingMaxwell's equations language

In these formulas e0 is the electric permittivity in vacuo; c is the electric susceptibility, the amplitude of the response to a unit field; w is the oscillation frequency of the harmonic

electromagnetic field; s is the frequency-dependent complex conductivity (for long wavelengths s is, to a good approximation, real); k the extinction coefficient

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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.

Homogeneous and isotropic medium

Harmonic wave equation

Complex harmonic field:

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Building an orthogonal vector wave basisfor the scattering on a sphere

P(R, , )q f

k

For a plane wave whose propagation vector is

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Mie’s spherical vector wave basis

The normalized Bessel functions describe the radial behavior.

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

TE

TE

TM

TM

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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

Notations for the boundary conditions

ki : incident wave-vector; it is chosen parallel with z-axis as usually in the experimental settingsks : scattered field wave-vector; the detector is placed in this direction ki and ks determine the scattering plane, which is highlighted q :scattering angle, in the scattering plane; in the the perpendicular plane f: azimuth angleThe fields have an in-plane transverse component Ep and a component Es perpendicular to the scattering plane.

Light scattering on a single sphere and standard decomposition in two perpendicular fields for measurements purposes

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Ashkin and Dziedzic proving Mie’s theory on a single levitated sphere

~2

~ 2z(x,x)y

z(x,z)x

There are two exceptional directions for which the secondary field is linearly polarized:•when f=0, Ef=0; if the detector is placed along the axis of incident plane wave polarization in the geometry z(x,z)x, only the component Eq is detected;•when = /f p 2, Eq=0; if the detector is placed perpendicular on the axis of incident plane wave polarization in the geometry z(x,x)y, only the component Ef is detected.

EOM: electro optic modulator controlling laser input powerS2, S3, S4 :screens for near-field viewA2, A3, A4: apertures for increased resolution in far-fieldAR: anti-reflection coating for avoiding interferences on D1P: tilted plate to avoid backscatteringD1,D2,D3,D4: detectorsMic1: projects particle imageMic2: collects backscattered lightMic3: collects the “p” scattered lightMic4: collects the “s” scattered lightFocusing lens: f= 60mm

Levitated silicone oil drop•radius a=11.4mm•refractive index N=1.4

P~15mW

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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

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~

~

• 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

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

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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.

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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. •mr, with a red line, is multiplied with a factor of ten for clarity;• mq=1=const is the green line; •ez=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.

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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 parameterr: radius of the corners

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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.

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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.

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

TE electric field

TM magnetic field

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Adjusting standard Mie’s equations for an inhomogeneous and anisotropic medium

radial equation

homogeneous sphere anisotropic cloak

wave vector for a homogeneousand isotropic medium inside the sphere

wave vector for the anisotropic cloak

and

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transverse equation

Adjusting standard Mie’s equations for an inhomogeneous and anisotropic medium

are the associated Legendre polynomials

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Perfect material behavior

et =e 0[R2/(R2−R1)]e r = et[(r−R1)/2r2] μt = μ0[R2/R2−R1]μr = μt[(r−R1)/2r2]

R1 = 0.50 l and R2 = l

anisotropic cloak

Ex field distribution and Poynting vectors due to an Ex polarized plane wave incidence

onto an ideal cloak

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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 Qscatt=0 the cloak is perfect.

Case III: et,norm=2; mt,norm=1/2

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Summary• 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