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Negative Refraction

Asem Ampoumogli1582542

RijksUniversiteit GroningenNanoscience TopMaster 2006 Symposium

Groningen June 2006

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• Refraction is one of the fundamental phenomena in the interaction of matter and radiation along with reflection, absorption and scattering.

• When a ray of light traveling through medium A (e.g. air) of refractive index n1

arrives at a medium B (e.g. glass) of refractive index n2, it is refracted at the

interface.

• All kinds of waves refract i.e. acoustic waves, seismic waves, even sea waves!

Refraction

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Index of refraction

r rn

• The refractive index n is defined by the equation:

• Where:

•εr is the relative electrical permittivity, defined as εr=ε/ε0

•μr is the relative magnetic permeability, defined as μr =μ/μ0

Both these constants are generally positive numbers.

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Index of refraction

1 1 2 2sin sinn n

• In 1621 the Dutch mathematician Willebrord Snell arrived at the sinθ form of the law of refraction.

p

cu

n

• Upon entering medium B, the radiation’s phase velocity will change to:

The refractive index is always positive.

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Veselago’s classic paperIn 1968 the Russian physicist Victor Veselago published a paper in which he studied the interaction of electromagnetic radiation with a (hypothetical) material for which both ε and μ are simultaneously negative.

He arrived at some interesting conclusions:

• A planar slab of this material will focus light (not parallel rays though).

• The vector set k, E, H will be left handed, which means the group and the phase velocities in this material will be in opposite directions.

•The energy flow E×H of an electromagnetic wave will be in the opposite direction to the wave vector because the permeability is negative.

V.G. Veselago, Soviet Physics USPEKHI, 10, 509 (1968).

http://sagar.physics.neu.

.edu/lhm-intro-1.html

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Veselago’s classic paper

Most importantly, Veselago was able to show that this material would have a negative index of refraction.

exp( ) exp( ) exp( ) 12 2

i in i

This has some more interesting consequences:

• The Doppler shift will be reversed.

• The Cerenkov radiation will be reversed.

• Snell’s law will be reversed.

How can this happen? What does it mean? Why are there no materials readily found that exhibit a negative refractive index?

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The Drude –Lorentz model and negative response

In the Drude-Lorentz atomic model, the atoms are modeled as oscillators (the electrons are bound to the nucleus with a spring) with a resonant frequency ω0.

An electric field of frequency ω will drive the oscillations of the system at ω.

If the frequency of the radiation is near the resonant frequency the induced polarization becomes very large.

In the classical analogue (mass on an ideal spring), below the resonant frequency, the mass is displaced in the same direction as the driving force (analogy: mass=dipole moment).

However, above the resonant frequency, the mass is displaced in a direction opposite to the driving force.

In the case of resonance with the electric field, the material will exhibit a negative permittivity ε (as is the case with metals at optical frequencies).

At resonance with a driving magnetic field, the material will exhibit a negative permeability .

So, the negative index of refraction will be observed when the material is in resonance with both the electric field and the magnetic field.

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Negative Index materialsCan we create materials that will have a negative index even for a narrow

frequency window?

In 1999 Professor J. Pendry of the London Imperial college published a paper proposing structures that, was predicted, would feature the needed electromagnetic properties to have a measurable negative refractive index.

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, IEEE transactions on microwave theory and techniques, 47, 2075 (1999).

The proposed structures are to be created with two elements:

Cut wires (designed to have a specific electrical resonance frequency) and

Split ring resonators (SRR’s) (designed to have a specific magnetic resonance frequency).

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Negative Index materialsBecause the radiation that will be used has a wavelength hundreds of times larger than the atomic units in our material, the atomic details lose importance in describing how the material will interact with it.

It was further argued that we can define an effective permittivity and an effective permeability to macroscopically describe our materials.

From this point of view we will have created a meta-material whose unique properties are not determined by the fundamental physical properties of its constituents but by the shape and the distribution of the specific patterns included in them.

What would something like that look like?

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Permittivity

2 20

2 20

( ) 1 peff i

The resonant frequency can be set to virtually any value in this kind of materials, so the negative ε can be reproduced at low frequencies rather than just the optical region.

Above ω0 and below ωp the effective

permittivity is negative.

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Permeability

The split ring resonator (SRR) will feature a magnetic response without being inherently

magnetic.

We will have the meta-material equivalent of a magnetic atom.

0 1/LC LC

2

2 20

( ) 1eff

F

i

12

Other designs

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Experimental verificationR. A. Shelby, D. R. Smith, & S. Schultz, Science 292, 77-79 (2001).

The experiment was carried out at a frequency of 10.5 GHZ.

The control sample was made of Teflon, cut with a step pattern identical to that of the LHM sample

Results:

NIM: n= - 2.7

Teflon: n= 1.4

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ApplicationsThe most prominent application of refractive materials is in the manufacturing of lenses.

Veselago (as well as Pendry) have shown that NIM lenses would not require curved surfaces to focus radiation (but would not focus parallel rays and will have a magnification of unity).

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ApplicationsOne of the most dramatic – and controversial- prediction for the NIMs was that by Pendry in 1999 which stated that a thin negative-index film should behave as a “superlens”, providing image detail with a resolution beyond the diffraction limit, to which all positive-index lenses are subject.

0

0 0

cos

sin 2z

x

k k

k k k

This restriction is a significant problem.

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ApplicationsMagnetic Resonance Imaging (MRI):

Large quasi-static fields cause the nuclear spins in the patient’s body to align. The spins are resonant at the local Larmor frequency, so a second magnetic field (an RF field) will excite them causing them to precess around the magnetic field. Images are constructed by observing the time dependent signal resulting from the precession of the spins.