Negative Refraction & the Perfect Lens

Home , Negative refraction, Victor Veselago

JB Pendry The Blackett Laboratory, Imperial College London http://www.cmth.ph.ic.ac.uk/photonics/

Some Reviews Some Popular Articles

Controlling Electromagnetic Fields The Quest for the superlens Science 312 1780-2 (2006). Scientific American 60- 67 July (2006). JB Pendry, D Schurig, and DR Smith Manipulating the near field with metamaterials Not Just a Light Story Optics & Photonics News 15 33-7 (2004) Nature Materials 5 755-64 (2006) Reversing Light with Negative Refraction Negative Refraction Physics Today 57 [6] 37-43 (June 2004) Contemporary Physics 45 191-202 (2004)

Metamaterials and Negative Refractive Index Science 305 788-92 (2004)

Refraction of Light – Snell/Descartes

θ1

θ2

Willebrord Snell van Roijen (or Snellius) (1580- 1626) The Snell-Descartes law of refraction: sin θ n = 1 René Descartes (1596 –1650) sin θ2 where n is the refractive index of the material

31 January 2010 page 2

12 March 2006 page 2 Focussing light

exp(ik00 x sinθ+ ik z cos θ− i ω t)

Galileo by Leoni - 1624 lens, n. L. lens lentil, from the similarity in form. A piece of glass with two curved surfaces

Maxwell’s Equations

∇×EB =−∂ ∂t ∇×HD =+∂ ∂t

∇⋅D = ρ ∇⋅B =0

James Clerk Maxwell (1831–1879)

Einstein, Light, and Geometry – the theory

The general theory of relativity: gravity changes geometry. Therefore gravity should bend light

19 February 2005 page 5

Negative Refractive Index and Snell’s Law

sin (θ1) n = () sin θ2 Hence in a negative refractive index material, light makes a negative angle with the normal.

This has unexpected consequences ……..

12 March 2006 page 3

A negative refractive index pool

The consequences of negative refraction 1. negative group velocity

energy flow and group velocity θ1 θ1

θ2 n =−1 rays wave wave velocity (energy flow) vectors

In a negative refractive index material, light Materials with negative refraction are sometimes makes a negative angle with the normal. Note that called left handed materials because the Poynting the parallel component of wave vector is always vector has the opposite sign to the wave vector. preserved in transmission, but that energy flow is opposite to the wave vector.

02 November 2003 page 11

Recipe for Negative Refractive Index James Clark Maxwell showed that light is an electromagnetic wave and its refraction is determined by both: the electrical permittivity, ε, and the magnetic permeability, μ.

The wave vector, k, is related to the frequency by the refractive index, −11− kcnc=εμω=ω00 Normally n, ε, and μ are positive numbers.

In 1968 Victor Veselago showed that if ε and μ are negative, we are forced by Maxwell’s equations to choose a negative square root for the refractive index,

n = −εμ,0,0 ε< μ<

20 February 2005 page 8 Negative Refraction - n < 0 µ where, ε<0 ε>0 kc= ω×εµ=ω× cn µ>0 µ>0 ni=+α n > 0 Either ε<0, or µ < 0, ensures that k opaque transparent is imaginary, and the material opaque. ε If ε < 0 and µ < 0, then k is real, but transparent, we are forced to choose the negative but different opaque square root to be consistent with ε<0 ε>0 Maxwell’s equations. µ<0 µ<0 n < 0 ni=+α' ε <µ<0, 0 means that n is negative

The wave vector defines how light propagates:

EE=−ω0 exp() ikzit

02 November 2003 page 9 What is a ‘metamaterial’ Conventional materials: properties derive from Metamaterials: properties derive from their their constituent atoms. constituent units. These units can be engineered as we please.

02 November 2003 page 2 Negative refraction: ε<0, µ< 0

Structure made at UCSD by David Smith Refraction of a Gaussian beam into a negative index medium. The angle of incidence is 30° (computer simulation by David Smith UCSD)

nini()ω=−+−+1.66 0.003 ,( ω=−+) 1.00 0.002 , ∆ωω= 0.07 Negative Refraction at the Phantom Works

Boeing PhantomWorks 32° wedges

Left: negatively refracting sample Right: teflon

Negative Refractive Index and Focussing

A negative refractive index medium bends light to a negative angle relative to the surface normal. Light formerly diverging from a point source is set in reverse and converges back to a point. Released from the medium the light reaches a focus for a second time.

05 December 2004 page 5 The consequences of negative refraction 3. Perfect Focussing

A conventional lens has resolution limited by the wavelength. The missing information resides in the near fields which are strongly localised near the object and cannot be focussed in the normal way.

The new lens based on negative refraction has unlimited resolution provided that the condition n =−1 is met exactly. This can happen only at one frequency. (Pendry 2000).

The secret of the new lens is that it can focus the near field and to do this it must amplify the highly localised near field to reproduce the correct amplitude at the image.

02 November 2003 page 19

Limitations to the Performance of a Lens Contributions of the far field to the image …..

exp(ik00 x sinθ+ ik z cos θ− i ω t)

….. are limited by the free space wavelength: θ=90 °gives maximum value of kkx ==ω=πλ00 c2 0 − the shortest wavelength component of the 2D image. Hence resolution is no better than, 22π πc Δ ≈= =λ0 k0 ω

20 February 2005 page 9

Fermat’s Principle:

e.g. for a lens the shortest optical distance between object and image is:

nd11+ n 2 d 2+=+ nd 13 nd 1''' 1 n 2 d 2 + nd 1 3 both paths converge at the same point because both correspond to a minimum. “Light takes the shortest optical path between two points”

19 February 2005 page 11

Fermat’s Principle for Negative Refraction

If n2 is negative the ray traverses negative optical space.

for a perfect lens (nn21=− ) the shortest optical distance between object and image is zero: 0 = nd++ n d nd 11 2 2 13 =+nd11''' n 2 d 2 + nd 13

For a perfect lens the image is the object

24 February 2005 page 12

Transformation optics & negative refraction The Veselago lens can be understood in terms of transformation optics if we allow ‘space’ to take on a negative quality i.e. space can double back on itself so that a given event exist on several manifolds:

The ‘Poor Man’s Superlens’

The original prescription for a superlens: a slab of material with

ε =−1, µ=− 1

However if all relevant dimensions (the thickness of the lens, the size of the object etcetera) are much less than the wavelength of light, electric and magnetic fields are decoupled. An object that comprises a pure electric field can be imaged using a material with,

ε =−1, µ=+ 1 because, in the absence of a magnetic field, µ is irrelevant.

We can achieve this with a slab of silver which has ε < 0 at optical frequencies.

16 January 2005 page 2 Anatomy of a Superlens The superlens works by resonant However, excitation of surface plasmons in the silver, wavefield “anti” surface surface plasmon of object plasmon wavefield wavefield

silver slab surface plasmon wavefield silver slab

Matching the fields at the boundaries At the same frequency as the surface selectively excites a surface plasmon plasmon there exists an unphysical on the far surface. “anti” surface plasmon - wrong boundary conditions at infinity,

“anti” surface plasmon wavefield silver slab

02 November 2003 page 20 Near field superlensing experiment: Nicholas Fang, Hyesog Lee, Cheng Sun and Xiang Zhan, UCB

Left: the objects to be imaged are inscribed onto the chrome. Left is an array of 60nm wide slots of 120nm pitch. The image is recorded in the photoresist placed on another side of silver superlens.

Below: Atomic force microscopy of a developed image. This clearly shows a superlens imaging of a 60 nm object (λ/6).

16 January 2005 page 3 Imaging by a Silver Superlens Nicholas Fang, Hyesog Lee, Cheng Sun, Xiang Zhang, Science 534 308 (2005)

(A) FIB image of the object. The line width of the ‘‘NANO’’ object was 40 nm. (B) AFM of the developed image on photoresist with a 35-nm-thick silver superlens. (C) AFM of the developed image on photoresist when the layer of silver was replaced by PMMA spacer as a control experiment. (D) blue line: averaged cross section of letter ‘‘A’’ line width 89nm red line: control experiment line width 321nm.

16 May 2005 page 12 Negative Space A slab of n =−1 material thickness d , cancels the effect of an equivalent thickness of free space. i.e. objects are focussed a distance 2d away. An alternative pair of complementary media, each cancelling the effect of the other. The light does not necessarily follow a straight line path in each medium:

General rule: two regions of space optically cancel if in each region ε,µ are reversed mirror images.

The overall effect is as if a section of space thickness 2d were removed from the experiment. A Negative Paradox

The left and right media in this 2D system are negative mirror images and therefore optically annihilate one another. However a ray ε→−1 construction appears to contradict µ→−1 this result. Nevertheless the theorem is correct and the ray construction erroneous. Note the closed loop of rays indicating the presence of resonances. ε=+1 µ=+1

2 1

Einstein, Light, and Geometry – the theory

The general theory of relativity: gravity changes geometry. Therefore gravity should bend light

Einstein, Eddington, Light, and Geometry – the experiment

In 1919 Sir Arthur Eddington observed a total solar eclipse and measured the deviation of starlight by the sun. His results were reported on the front page of The Times and changed science forever. Controlling Electromagnetic Fields Exploiting the freedom of design which metamaterials provide, we show how electromagnetic fields can be redirected at will and propose a design strategy. The conserved fields: electric displacement field, D, magnetic induction field, B, and Poynting vector, S, are all displaced in a consistent manner and can be arranged at will by a suitable choice of metamaterials.In general we require materials that are anisotropic and spatially dispersive.

Left: a field line in free space with the background Cartesian coordinate grid shown. Right: the distorted field line with the background coordinates distorted in the same fashion. A Perfect Magnifying Glass

r2 ε=ε=ε=+2 ,0 <

µ=εxx,, µ=ε yy µ=ε zz

It is possible to design a spherical annulus of negative material lying between r2 and r3 that acts like a magnifying glass. To the outside world the contents of the sphere radius r3 appear to fill the larger sphere radius r1 with proportionate magnification. ` 1a) 1b) An optical turbine. A plane wave entering the red sphere from the left is captured and compressed inside the green sphere. a) A ray picture which shows only part of the rays being captured b) An exact solution of Maxwell’s equations. The green sphere is filled with the compressed contents of the red sphere as predicted. The region outside the blue sphere is free space.

Transformation Optics Shrinks Optical Devices Wei Hsiung Wee, New Journal of Physics 11 (2009) 073033 Example 1: a shadow created by a sub wavelength device left: loss tangent of 510× −3, right: lossless system

Yellow inner cylinder: perfectly absorbing material blue annulus: magnifying superlens – radius about ≈ λ see also: Yang T, Chen H, Luo X and Ma H 2008 Opt. Express 16 18545–50

Transformation Optics Shrinks Optical Devices Wei Hsiung Wee, New Journal of Physics 11 (2009) 073033 Example 2: a retro-reflector created by a sub wavelength device

Contour plot of the retro reflected field intensity from a super-scatterer containing a retro-reflecting cylinder. In this example the effect is to double the size of the retro-reflector.

Negative Refraction & the Perfect Lens

JB Pendry The Blackett Laboratory, Imperial College London http://www.cmth.ph.ic.ac.uk/photonics/

Some Reviews Some Popular Articles

Metamaterials and Negative Refractive Index Science 305 788-92 (2004)