Negative Refraction & the Perfect Lens

Negative Refraction & the Perfect Lens JB Pendry The Blackett Laboratory, Imperial College London http://www.cmth.ph.ic.ac.uk/photonics/ Some Reviews Controlling Electromagnetic Fields Science 312 1780-2 (2006). JB Pendry, D Schurig, and DR Smith Not Just a Light Story Nature Materials 5 755-64 (2006) Negative Refraction Contemporary Physics 45 191-202 (2004) Some Popular Articles The Quest for the superlens Scientific American 60-67 July (2006). Manipulating the near field with metamaterials Optics & Photonics News 15 33-7 (2004) Reversing Light with Negative Refraction Physics Today 57 [6] 37-43 (June 2004) Metamaterials and Negative Refractive Index Science 305 788-92 (2004)

31 January 2010 page 2 Refraction of Light Snell/Descartes θ 1 θ 2 The Snell-Descartes law of refraction: sin θ n = 1 sin θ2 where n is the refractive index of the material Willebrord Snell van Roijen (or Snellius) (1580-1626) René Descartes (1596 1650)

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Focussing light ( ik x θ+ ik z θ iω t) exp sin cos 0 0 θ 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 E= B t H=+ D t D = B = ρ 0 James Clerk Maxwell (1831 1879)

Einstein, Light, and Geometry The general theory of relativity: gravity changes geometry. Therefore gravity should bend light the theory

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12 March 2006 page 3 Negative Refractive Index and Snell s Law ( θ1 ) ( θ ) sin n = sin 2 Hence in a negative refractive index material, light makes a negative angle with the normal. This has unexpected consequences..

A negative refractive index pool

02 November 2003 page 11 The consequences of negative refraction 1. negative group velocity θ 1 θ 1 energy flow and group velocity θ 2 n = 1 rays (energy flow) wave vectors wave velocity In a negative refractive index material, light makes a negative angle with the normal. Note that the parallel component of wave vector is always preserved in transmission, but that energy flow is opposite to the wave vector. Materials with negative refraction are sometimes called left handed materials because the Poynting vector has the opposite sign to the wave vector.

20 February 2005 page 8 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, 1 1 0 0 k = εμω c = nω c 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, ε< 0, μ< 0

02 November 2003 page 9 opaque ε< 0 µ>00 n=+α i transparent, but different µ Negative Refraction - n < 0 ε> 0 µ>00 n > 0 ε< 0 ε> 0 µ< 0 µ< 0 n < 0 n=+α i ' transparent opaque The wave vector defines how light propagates: E = E0 exp( ikz iω t) ε where, k = ω c εµ =ω c n Either ε< 0, or µ < 0, ensures that k is imaginary, and the material opaque. If ε < 0 and µ < 0, then k is real, but we are forced to choose the negative square root to be consistent with Maxwell s equations. ε < 0, µ< 0 means that n is negative

02 November 2003 page 2 Conventional materials: properties derive from their constituent atoms. What is a metamaterial Metamaterials: properties derive from their constituent units. These units can be engineered as we please.

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) ( ) ( ) n ω = 1.66 + 0.003 i, n ω = 1.00 + 0.002 i, ω ω= 0.07 +

Negative Refraction at the Phantom Works

Boeing PhantomWorks 32 wedges Left: negatively refracting sample Right: teflon

05 December 2004 page 5 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.

02 November 2003 page 19 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.

20 February 2005 page 9 Limitations to the Performance of a Lens Contributions of the far field to the image.. ( ik x θ+ ik z θ iω t) exp sin cos 0 0 θ.. are limited by the free space wavelength: θ= 90 gives maximum value of kx = k0 =ω c0 = 2π λ 0 the shortest wavelength component of the 2D image. Hence resolution is no better than, 2π 2πc Δ = =λ0 k ω 0

19 February 2005 page 11 Fermat s Principle: Light takes the shortest optical path between two points e.g. for a lens the shortest optical distance between object and image is: nd + n d + nd = nd' + n d' + nd' 1 1 2 2 1 3 1 1 2 2 1 3 both paths converge at the same point because both correspond to a minimum.

24 February 2005 page 12 Fermat s Principle for Negative Refraction If n 2 is negative the ray traverses negative optical space. for a perfect lens ( n 2 = n 1 ) the shortest optical distance between object and image is zero: 0 = nd + n d + nd 1 1 2 2 1 3 = nd' + n d' + nd' 1 1 2 2 1 3 For a perfect lens the image is the object

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:

16 January 2005 page 2 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.

02 November 2003 page 20 The superlens works by resonant excitation of surface plasmons in the silver, Anatomy of a Superlens However, wavefield of object anti surface plasmon wavefield surface plasmon wavefield silver slab surface plasmon wavefield silver slab At the same frequency as the surface plasmon there exists an unphysical anti surface plasmon - wrong boundary conditions at infinity, Matching the fields at the boundaries selectively excites a surface plasmon on the far surface. anti surface plasmon wavefield silver slab

16 January 2005 page 3 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 May 2005 page 12 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.

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 ε 1 µ 11 ε=+ 1 µ=+1 1 The left and right media in this 2D system are negative mirror images and therefore optically annihilate one another. However a ray construction appears to contradict this result. Nevertheless the theorem is correct and the ray construction erroneous. Note the closed loop of rays indicating the presence of resonances. 2 1

Einstein, Light, and Geometry The general theory of relativity: gravity changes geometry. Therefore gravity should bend light the theory

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 r 3 r 1 3 r 1 r 2 2 2 2 3 2 r2 2 r ε x =ε y =ε z =+, 0< r < r r ε x =ε y =εz, r < r < r r ε =ε =ε =+ 1, r < r < x y z µ =ε, µ =ε, µ =ε x x y y z z 2 3 3 2 It is possible to design a spherical annulus of negative material lying between r 2 and r 3 that acts like a magnifying glass. To the outside world the contents of the sphere radius r 3 appear to fill the larger sphere radius r 1 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 5 10 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.