Bernd Hüttner DLR Stuttgart Folie 1 A journey through a strange classical optical world Bernd Hüttner CPhys FInstP Institute of Technical Physics DLR Stuttgart Left-handed media Metamaterials Negative refractive index

Bernd Hüttner DLR Stuttgart Folie 2 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plasmon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 3 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 4 A short historical background V G Veselago, "The electrodynamics of substances with simultaneously negative values of eps and mu", Usp. Fiz. Nauk 92, (1967) A Schuster in his book An Introduction to the Theory of Optics (Edward Arnold, London, 1904). J B Pendry „Negative Refraction Makes a Perfect Lens” PHYSICAL REVIEW LETTERS 85 (2000) H Lamb (1904), H C Pocklington (1905), G D Malyuzhinets, (1951), D V Sivukhin, (1957); R Zengerle (1980)

Bernd Hüttner DLR Stuttgart Folie 5 Objections raised against the topic 1.Valanju et al. – PRL 88 (2002) Wave Refraction in Negative- Index Media: Always Positive and Very Inhomogeneous 2. G W 't Hooft – PRL 87 (2001) Comment on “Negative Refraction Makes a Perfect Lens” 3. C M Williams - arXiv:physics (2001) - Some Problems with Negative Refraction

Bernd Hüttner DLR Stuttgart Folie 6 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 7

Bernd Hüttner DLR Stuttgart Folie 8 Photonic crystals

Bernd Hüttner DLR Stuttgart Folie 9 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 10 Left-handed metamaterials (LHMs) are composite materials with effective electrical permittivity, ε, and magnetic permeability, µ, both negative over a common frequency band. Definition: What is changed in electrodynamics due to these properties? Taking plane monochromatic fields Maxwell‘s equations read Note, the changed signs

Bernd Hüttner DLR Stuttgart Folie 11 By the standard procedure we get for the wave equation no change between LHS and RHS Poynting vector

Bernd Hüttner DLR Stuttgart Folie 12 RHS LHS

Bernd Hüttner DLR Stuttgart Folie 13 Two (strange) consequences for LHM

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Bernd Hüttner DLR Stuttgart Folie 15 Why is n < 0? 1. Simple explanation 2. A physical consideration 2 nd order Maxwell equation: 1 st order Maxwell equation: RHS:  > 0,  > 0, n > 0LHS:  < 0,  < 0, n < 0

Bernd Hüttner DLR Stuttgart Folie 16 whole parameter space

Bernd Hüttner DLR Stuttgart Folie 17 The averaged density of the electromagnetic energy is defined by Note the derivatives has to be positive since the energy must be positive and therefore LHS possess in any case dispersion and via KKR absorption 3. An other physical consideration

Bernd Hüttner DLR Stuttgart Folie 18 Kramers-Kronig relation Titchmarsh‘theorem: KKR causality

Bernd Hüttner DLR Stuttgart Folie 19 Because the energy is transported with the group velocity we find This may be rewritten as Since the denominator is positive the group velocity is parallel to the Poynting vector and antiparallel to the wave vector.

Bernd Hüttner DLR Stuttgart Folie 20 The group velocity, however, is also given by We see n < 0 for vanishing dispersion of n This should be not confused with the superluminal, subluminal or negative velocity of light in RHS. These effects result exclusively from the dispersion of n.

Bernd Hüttner DLR Stuttgart Folie 21 Dispersion of ,  and n Lorentz-model

Bernd Hüttner DLR Stuttgart Folie 22 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 23 Reflection and refraction but what is with µ = 1 Optically speaking a slab of space with thickness 2W is removed. Optical way is zero !

Bernd Hüttner DLR Stuttgart Folie 24 Snellius law for LHS Due to homogeneity in space we have k 0x = k 1x = k 2x

Bernd Hüttner DLR Stuttgart Folie 25 water: n = 1.3„negative“ water: n = -1.3 First example

Bernd Hüttner DLR Stuttgart Folie 26  = 2.6 left-measured right-calculated  = -1.4 left-measured right-calculated Second example: real part of electric field of a wedge

Bernd Hüttner DLR Stuttgart Folie 27 General expression for the reflection and transmission The geometry of the problem is plotted in the figure where r 1 ’ = -r 1.

Bernd Hüttner DLR Stuttgart Folie 28 e 1 =  1 =1, e 2 = m 2 = -1 and u 0 = 0 we get R = 0 & T = 1 1. s-polarized

Bernd Hüttner DLR Stuttgart Folie p-polarized R = 0 – why and what does this mean? Impedance of free space Impedance for e = m = -1 invisible!

Bernd Hüttner DLR Stuttgart Folie 30 Reflectivity of s-polarized beam of one film

Bernd Hüttner DLR Stuttgart Folie 31 Absorption or reflection of a normal system

Bernd Hüttner DLR Stuttgart Folie 32 Reflection of a normal system

Bernd Hüttner DLR Stuttgart Folie 33 Reflection of a LHS

Bernd Hüttner DLR Stuttgart Folie 34 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 35 Invisibility Al plate, d=17µm

Bernd Hüttner DLR Stuttgart Folie 36 An other miracle: Cloaking of a field For the cylindrical lens, cloaking occurs for distances r 0 less than r # if  c =  m The animation shows a coated cylinder with  in =1,  s =-1+i·10 -7, r out =4, r in =2 placed in a uniform electric field. A polarizable molecule moves from the right. The dashed line marks the circle r=r #. The polarizable molecule has a strong induced dipole moment and perturbs the field around the coated cylinder strongly. It then enters the cloaking region, and it and the coated cylinder do not perturb the external field.

Bernd Hüttner DLR Stuttgart Folie 37 There is more behind the curtain: 1. outside the film Due to amplification of the evanescent waves perfect lens – beating the diffraction limit How can this happen? Let the wave propagate in the z-direction the larger k x and k y the better the resolution but k z becomes imaginary if How does negative slab avoid this limit?

Bernd Hüttner DLR Stuttgart Folie 38 Amplification of evanescent waves

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Bernd Hüttner DLR Stuttgart Folie 40 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 41 How can we understand this? Analogy – enhanced transmission through perforated metallic films Ag d=280nm hole diameter d / = 0.35 L=750nm hole distant area of holes 11% h =320nm thickness d opt =11nm optical depth T film ~ solid film

Bernd Hüttner DLR Stuttgart Folie 42 Detailed analysis shows it is a resonance phenomenon with the surface plasmon mode. Surface-plasmon condition:

Bernd Hüttner DLR Stuttgart Folie 43 Interplay of plasma surface modes and cavity modes The animation shows how the primarily CM mode at 0.302eV (excited by a normal incident TM polarized plane wave) in the lamellar grating structure with h=1.25μm, evolves into a primarily SP mode at 0.354eV when the contact thickness is reduced to h=0.6μm along with the resulting affect on the enhanced transmission.

Bernd Hüttner DLR Stuttgart Folie 44 Beyond the diffraction limit: Plane with two slits of width /20  =1  =2.2  =-1 µ=-1  =-1+i·10 -3 µ=-1+i·10 -3

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Bernd Hüttner DLR Stuttgart Folie 46 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 47 There is more behind the curtain: 2. inside the film The peak starts at the exit before it arrives the entry Example. Pulse propagation for n = -0.5 Oje, is this mad?!No, it isn’t!

Bernd Hüttner DLR Stuttgart Folie 48 An explanation: Let us define the rephasing length l of the medium where v g is the group velocity Remember, Fourier components in same phase interfere constructively If the rephasing length is zero then the waves are in phase at  =  0

Bernd Hüttner DLR Stuttgart Folie 49 RHS LHS RHS Peak is at z=0 at t=0 t < 0 the rephasing length l II inside the medium becomes zero at a position z 0 = ct / n g. At z 0 the relative phase difference between different Fourier components vanishes and a peak of the pulse is reproduced due to constructive interference and localized near the exit point of the medium such that 0 > t > n g L/c. The exit pulse is formed long before the peak of the pulse enters the medium

Bernd Hüttner DLR Stuttgart Folie 50 At a later time t’ such that 0 > t’ > t, the position of the rephasing point inside the medium z 0 ’ = ct’/n g decreases i.e., z 0 ’ < z 0 and hence the peak moves with negative velocity -v g inside the medium. t=0: peaks meet at z=0 and interfere destructively. Region 3: since 0 >t>n g L/c is z 0 ’’ > L 0>t’>t: z 0 ’’’ > z 0 ’’ the peak moves forward

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Bernd Hüttner DLR Stuttgart Folie 52 Gold plates (300nm) and stripes (100nm) on glass and MgF 2 as spacer layer

Bernd Hüttner DLR Stuttgart Folie 53 Overview 1. Short historical background 2. What are metamaterials? 3. Electrodynamics of metamaterials 4. Optical properties of metamaterials 5. Invisibility, cloaking, perfect lens 6. Surface plamon waves and other waves 7. Faster than light 8. Summary

Bernd Hüttner DLR Stuttgart Folie 54 Summary Metamaterials have new properties: 1. S and v g are antiparallel to k and v p 2. Angle of refraction is opposite to the angle of incidence 3. A slab acts like a lens. The optical way is zero 4. Make perfect lenses, R = 0, T = 1 5. Make bodies invisible 6. Can be tuned in many ways

Bernd Hüttner DLR Stuttgart Folie 55 n W = 1.35 n G = 1.5 n W = 1.35 n G = -1.5 n W = n G = 1.5 n W = n G = -1.5