1. RijksUniversiteit Groningen Nanoscience TopMaster 2006 Symposium
Negative Refraction
Asem Ampoumogli
Groningen June 2006

2. Refraction
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!
Sea waves: when they feel the sea bottom
Refraction creates rainbows and makes lenses work

3. Index of refraction 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
The permittivity ε is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field.
Permittivity relates therefore to a material's ability to transmit (or "permit") an electric field. D is the electric displacement, E is the electric field
Permeability is the degree of magnetization (the amount of magnetic moment per unit volume) of a material that responds linearly to an applied magnetic field. B is the magnetic flux, H is the magnetic induction field (auxiliary magn. Field)
Both these constants are generally positive numbers. Negative values are not forbidden but are rare, metals have a negative permittivity in optical frequencies
Both these constants are generally positive numbers.

4. Index of refraction
In 1621 the Dutch mathematician Willebrord Snell arrived at the sinθ form of the law of refraction.
Upon entering medium B, the radiation’s phase velocity will change to:
Descartes also arrived at this law in 1637 – in france it is called Descarte’s law or Snell-Descartes law
a small portion of each incident angled wavefront (angled relative to the interface) should impact the second medium before the rest of the front reaches the interface (Fig. 2). This portion will start to move through the second medium while the rest of the wave is still traveling in the first medium, but will move more slowly due to the higher refractive index of the second medium. Because the wavefront is now traveling at two different speeds, it will bend into the second medium, thus changing the angle of propagation.
The refractive index is always positive and in nearly all cases larger than 1
The refractive index is always positive.

5. Veselago’s classic paper
In 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.
V.G. Veselago, Soviet Physics USPEKHI, 10, 509 (1968).
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.
Termed these materials left handed – a term still used in literature today.
Maxwell’s electric field curl eqn: ΔχΕ=-θΒ/θτ
.edu/lhm-intro-1.html

6. Veselago’s classic paper
Most importantly, Veselago was able to show that this material would have a negative index of refraction.
This has some more interesting consequences:
The Doppler shift will be reversed.
The Cerenkov radiation will be reversed.
Snell’s law will be reversed.
Doppler:the freqency of a light source moving toward an observer being down-shifted in frequency
Cerenkov: Cerenkov radiation will emerge in the opposite direction to the direction of the incident particle
Snell’s law: in the denser medium a light ray will be refracted away from the normal
How can this happen? What does it mean? Why are there no materials readily found that exhibit a negative refractive index?

7. 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 .
We turn for help to the drude-lorentz atomic model
2nd parag:
If the freq is below ω0 an applied electric field will induce a polarization in the same direction as the field.
if the frequency of the radiation is ω0 then we will have a resonance phenomenon.
4th paragraph
As the driving frequency, the radiation frequency, is swept through the resonance, the polarization flips from in-phase to out-of-phase. This is the negative response.
Mass: dipole moment
The negative resonance response translates to a negative material response
This helps us understand why there are no NI materials in nature:
--The resonances in existing materials that give rise to electric polarizations typically occur at very high frequencies (in the optical region for metals or at least in the THz to IR region for semiconductors and insulators
---On the other hand, resonances in magnetic systems typically occur at much lower frequencies (usually tailing off toward the THz and infrared region) plus generally occur in inherently magnetic materials (typically include ferromagnetic or antiferromagnetic systems)
and result from phonon modes, plasma like oscillations of the conduction electrons, or other fundamental processes).
So, the negative index of refraction will be observed when the material is in resonance with both the electric field and the magnetic field.

8. Negative Index materials
Can 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).
Frequency dispersion: they will vary as a function of radiation frequency
The negative material parameters will occur near the resonance. This has two implications:
Negative material parameters will exhibit frequency dispersion.
The usable bandwidth will be relatively narrow.
--in the microwave region

9. Negative Index materials
Because 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.
Cut wires: wires into which cuts have been periodically introduced
In practice we can average over the atomic scale by (conceptually) replacing the otherwise inhomogeneous medium by a homogeneous material characterized just by two macroscopic electromagnetic parameters, the permittivity and the permeability.
From the electromagnetic point of view the wavelength λ determines whether a collection of atoms or other objects can be considered a material [8].
The electromagnetic parameters need not strictly arise from the response of atoms or molecules: Any collection of objects whose size and spacing are much smaller than λ can be described by an ε and μ.
What would something like that look like?

10. Permittivity
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.
A composite left-handed medium. The medium consists of metallic posts and Split Ring Resonators (SRR’s) created lithographically on a circuit board. (B) A split ring structure etched into circuit boards with Cu wires. This material gives negative electric and magnetic response
where the plasma frequency ωp and the resonance frequency ω0 are determined only by the geometry of the lattice rather than by the charge, effective mass and density of electrons, as is the case in natural materials. At frequencies above ω0 and below ωp the permittivity is negative.
Above ω0 and below ωp the effective permittivity is negative.

11. We will have the meta-material equivalent of a magnetic atom.
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.
It can be calculated that a magnetic response (a magnetic dipole) can be obtained By circulating currents in a closed SRR (CSRR). we get a magnetic moment m with magnitude given by the product of the current and the area of the CSRR and direction perpendicular to the plane of the ring. Therefore the CSRR behaves as an inductor. I fit is coupled with a capacitor (realized by making a cut) then we get an LC circuit – this will produce resonant magnetization
This is the metamaterial equivalent of a magnetic atom
The Boeing Cube – negative index at GHz frequencies
Although such an inhomogeneous collection may not satisfy our intuitive definition of a material, an EM wave passing through cannot tell the difference
Scaling these structures down we increase the frequency where we will get a resonant response – a negative index of refraction

12. Other designs
LEFT: (A) Schematic representation of one unit cell of the wire-pair structure.
(B) Photograph of fabricated microwave-scale wire-pair sample.
RIGHT: A): photo of the solid state sample. Applying a standard hot-press technique for the manufacture of printed circuit boards, the researchers have created a solid-state metamaterial sample of the left-handed medium by compressing pieces of alternately stacked PC boards with and without the Ω patterns - These are supposed to have very low losses

13. Experimental verification
R. A. Shelby, D. R. Smith, & S. Schultz, Science 292, (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
Fig. 1. Photograph of the lefthanded metamaterial (LHM) sample. The LHM sample consists of square copper split ring resonators and copper wire strips on fiber glass circuit board material. The rings and wires are on opposite sides of the boards, and the boards have been cut and assembled into an interlocking lattice.
--The detector was rotated around the circumference of the circle in 1.5¡ steps, and the transmitted power spectrum was measured as a function of angle θ from the interface normal (microwaves)

14. Applications
The 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).
1st: a material of index 1 has no refractive power – a material of -1 does!
For this lens, diverging rays from a nearby object are negatively refracted on the first interface (the left side of Fig. 10) reversing their trajectory so as to converge at a focus within the material. The rays diverge from this focus and are again negatively refracted at the second interface, finally converging to form a second image just outside the slab.
Pic: does not focus parallel rays – has a magnification of 1.

15. This restriction is a significant problem.
Applications
One 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.
Each ray emanating from an object has wave vector components along the axis of the lens kz=kocosθ , and perpendicular to the axis kx=k0sinθ . The former component is responsible for transporting the light from object to image and the latter represents a Fourier component of the image for resolution: the larger we can make kx, the better. The best result that can be achieved is k0 and hence the limit to resolution of Δ=..
This restriction is a significant problem in many areas of optics. The feature size that can be achieved lithographically is limited by the wavelength, as is the storage capacity of optical media
This trick of including the high-resolution but rapidly decaying part of the image is achieved by resonant amplification of the fields. Materials with either negative permittivity or negative permeability support a host of surface modes closely related to surface plasmons, commonly observed at metal surfaces (6), and it is these states that are resonantly excited. By amplifying the decaying fields of a source, the surface modes restore them to the correct amplitude in the image plane.
Unlike any other lens, the resolution limit of the planar negative-index lens is determined by how many evanescent waves from the object can be recovered, rather by the diffraction limit (in practice several stringent requirements limit the perfect focusing).
As a first step the weak incident near field excites surface plasmons on the first surface of the slab. Next the surface plasmons excited on the first surface begin a resonant interaction with the surface plasmons on the second surface and because the resonance is very strong the image is transferred to the second surface in a hugely amplified form. Τhe various components have been deliberately over-amplified so that the field escaping from the far side of the slab decays by just the right amount to bring all the components to exactly the right amplitude in the image plane
This restriction is a significant problem.

16. Applications Magnetic 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.
1st Any material destined for use in the MRI environment must not perturb the quasi-static magnetic field pattern, thus excluding the use of all conventional magnetic materials. However magnetic metamaterials that respond to time-varying fields but not to static fields can be used to alter and focus the RF fields without interfering with the quasi-static field pattern
Very expensive equipment – to buy and maintain but irreplaceable as a diagnostic tool
Primary application of NIM for many groups.
A sheet of metal coiled into a ‘Swiss roll’ responds to a magnetic field with a current. An array of
Swiss rolls produces a material which has negative magnetic permeability over a range of
Frequencies.
A single element of Swiss roll metamaterial. (B) An array of such elements is assembled into a slab and the RF magnetic field from an M-shaped antenna, placed below the slab, is reproduced on the upper surface. The red circles show the location of the rolls, which were 1 cm in diameter.
Other impressive applications involve a potential “cloaking device” that has very recently been described [24] by researchers at the Imperial College London. The idea consists of steering light around an object rendering it invisible. The details of the report are very sketchy at the time as the paper has not been made available at the time of writing, but it has become clear that the new material (which is described as being wrapped around the target object) will be functional across a large part of the electromagnetic spectrum.