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Photonic Crystals and Metamaterials and accelerator applications Rosa Letizia Lancaster University/ Cockcroft Institute [email protected] Cockcroft Institute, Spring term, 16/04/12 Lectures outline Lecture 1 – Introduction to Photonic Crystals • What is a photonic crystal • Bandgap property • Intentional defects in Photonic Crystals • Photonic Crystal applications examples Lecture 2 – Metamaterials • What is a metamaterial • Effective parameters from periodic unit cells • Permittivity and permeability models • Retrivial technique • Modelling and characterisation of metamaterials Lecture 3 – Accelerator applications • Photonic crystal resonant cavities • Dielectric laser accelerators • Metamaterials accelerating waveguides Lecture 4 – Computational Photonics • Introduction to Finite Difference Time Domain • Computational challenges Cockcroft Institute, Spring term, 16/04/12 Advanced functional materials Predesigned electromagnetic properties • Overcoming the limitations of natural materials by means of “function through structure” concept – Photonic Crystals technology – Metamaterials • Engineering of the geometry of the structure allows for creation of “artificial materials” for unusual EM responses • Scalability • Interference lithography (IL) holds the promise of fabricating large-area, defect-free 3D structures on the sub-micrometer scale both rapidly and cheaply Cockcroft Institute, Spring term, 16/04/12 What are Photonic crystals (PhCs)? 1-D 2-D 3-D PhC PhC PhC • Electronic crystal – a familiar analogy • a periodic array of atoms forms a lattice • lattice arrangement defines energy bands • The OPTICAL ANALOGY – Photonic Band Gap (PBG) crystal • a periodic array of optical materials forms a lattice • allowed energy (wavelength) bands arise atoms in diamond structure Cockcroft Institute, Spring term, 16/04/12 dielectric spheres, diamond lattice PhCs exist in nature Iridescence from butterfly wing The colours produced are not the results of the presence of pigments. The mixing of photonic structures and organic pigments will vary the shades we see [ P. Vukosic et al., Proc. Roy. Soc: Bio. Sci. 266, 1403 (1999) ] 3µm Cockcroft Institute, Spring term, 16/04/12 [ also: B. Gralak et al., Opt. Express 9, 567 (2001) ] Bragg’s diffraction law The condition that defines constructive interference: Interference between two diffracted waves from a series of atomic planes separated by d. m 2 d sin m 1,2,3.. Cockcroft Institute, Spring term, 16/04/12 Maxwell’s equations in periodic media First studied in 1982 by Bloch who extended a theorem developed by Floquet in 1883 for the 1D case. BLOCH’S THEOREM: waves in a periodic material can propagate with no scattering and their behaviour is ruled by a periodic envelop function which is multiplied by a plane wave. (for most λ, scattering cancels coherently) B E t D H t D 0 B 0 D 0 r E B 0 H 1 2 H H c r r a H n,k r e jkr un,k r Plane wave Has the periodicity of the crystal lattice BLOCH MODES Cockcroft Institute, Spring term, 16/04/12 Maxwell’s equations in periodic media • u n, k r is given by a finite unit cell so ωn(k) is discrete (the dispersion relation is organised in bands defined by the index n) •The solutions of the wave equation: • The inverse of the dielectric constant and the Bloch modes are expanded in Fourier series upon the reciprocal vector of the lattice, G r 1 G e jk r r G G G G ' a k a Brillouin Zone H n , k r uGn , k e j ( k G )r k G' k G u n, k G G Cockcroft Institute, Spring term, 16/04/12 n k 2 c2 uGn,'k , Origin of the bandgap A complete photonic bandgap is a range of frequencies ω in which there are no propagation solutions (real k) of Maxwell’s equations for any vector k and it is surrounded by propagation states above and below the forbidden gap x x a k ck a=0 usual dispersion relation. States can be defined as Bloch functions and wavevectors the bands for k>π/a are translated ‘folded’ into the first Brillouin zone Cockcroft Institute, Spring term, 16/04/12 The degeneracy is broken, the shift of the bands leads to the formation of bandgap A 1D PhC – Quarter-wave stack •All 1D periodicity in space give rise to a bandgap for any contrast of the refractive index. •Smaller the contrast and smaller is the size of the badgap •A peculiar case: quarter-wave stack can maximise the size of its photonic bandgap by making the all reflected waves from the layers exactly in phase one with each others at the midgap frequency Cockcroft Institute, Spring term, 16/04/12 2D Photonic Crystals Rectangular lattice Hexagonal lattice Cockcroft Institute, Spring term, 16/04/12 2D Photonic Crystals Cockcroft Institute, Spring term, 16/04/12 PhC and defects Breaking the periodicity: Point defect Cavity Joannopoupos, Photonic Crystals Molding the flow of light: jdj.mit.edu/book Cockcroft Institute, Spring term, 16/04/12 PhC and defects Line defect Waveguide Cockcroft Institute, Spring term, 16/04/12 PhC and defects BENEFITS • EM fields highly confined in defect regions • enhanced interaction EM excitation – structure (e. g. enhanced nonlinear effects for frequency conversion applications) • Scaled-down optical devices • Integration of multiple optical functionalities on single platform (all-optical circuits) Cockcroft Institute, Spring term, 16/04/12 PhC fibers • Wide single mode wavelength range • large effective mode area • anomalous dispersion at visible and near IR wavelengths • Hollow core is allowed (no limited by material absorption) Cockcroft Institute, Spring term, 16/04/12 [ R. F. Cregan et al., Science 285, 1537 (1999) ] [ B. Temelkuran et al., Nature 420, 650 (2002) ] 3D PhC – Woodpile structure Unit cell: 4 layers Cockcroft Institute, Spring term, 16/04/12 PhC applications PhC-based optical filter* *R. Letizia, and S. S. A. Obayya, IET Optoelectronics, vol. 2, n. 6, pp. 241253, 2008. Cockcroft Institute, Spring term, 16/04/12 PhC applications Photonic wire (PhW) y x w = 0.38 m a = 0.3162 m r = 0.23a nnl = 1.43 x 10-17 m2/W sat = 0.31 D Pinto, and S S A Obayya, IET Optoelectron., vol. 2, no. 6, pp. 254261, 2008 Cockcroft Institute, Spring term, 16/04/12 c=a n1 = 1.0 n2 = 3.48 a1 = 262 nm r1 = 51.93 nm a2 = 280 nm r 2= 63.94 nm PhC applications Linear res = 1.536 m Q = 427 T = 0.66 Nonlinear res = 1.543 m Q = 424 T = 0.52 Cockcroft Institute, Spring term, 16/04/12 PhC applications Enhanced Second Harmonic Generation Fundamental frequency (TM) SH frequency Cockcroft Institute, Spring term, 16/04/12 SH frequency (TE) Metamaterials (MTMs) • Control the flow of EM wave in unprecedented way • The design relies on inclusions and the new properties emerge due the specific interactions with EM fields • These designs can be scaled down and the MTM really behaves as a effectively continuous medium • Composite by elements as materials are composite of atoms • METAMATERIAL represents the “next” level of organisation of the matter the prefix “META” originates from Greek work “µετα” which means “beyond” Cockcroft Institute, Spring term, 16/04/12 MTM applications: where we are Industrial Applications • Information and Communication technologies • Space & Security and defence • Health • Energy • Environmental Device already realised • Sensors • Superlensing • Cloaking • Light Emitting Diodes/ cavities for low threshold Lasers Cockcroft Institute, Spring term, 16/04/12 MTM for cloaking Metamaterials are also currently a basis for building a cloaking device. A possibility of a working invisibility cloak was demonstrated in 2006 [1]. Following this result, an intense research work has been spent in order to build a cloaking device at optical frequencies [2 - 4]. [1] D. Schurig, et al., Science, 314, 977-980, 2006. [2] A. Greenleaf, et al., Phys. Rev. Lett., 102, 183901 (1-4), 2007. [3] X. Zhang, et al., Opt. Express, 16, 11764-11768, 2008. [4] R. Liu, et al., Science, 323, 366-369, 2009. Cockcroft Institute, Spring term, 16/04/12 Important concepts Backward-wave materials •Negative refractive index materials do not exist in nature. These type of materials were first theoretically introduced by Veselago but only in 90’s Pendry showed how physically realise them. •Backward wave media are materials in which the energy velocity direction is opposite to the phase velocity direction. In particular this takes place in isotropic materials with negative permittivity and permeability (double negative materials DNM). Negative refraction takes place at the interface between “normal” media and DNM Cockcroft Institute, Spring term, 16/04/12 Negative refractive index r j r r j r MTMs obey Snell’s law: n r r r r n1 sin 1 n2 sin 2 2 0 Cockcroft Institute, Spring term, 16/04/12 Superlens •Negative refraction can be used to focus light. A flat slab of material will produce two focal points, one inside the slab and the other one outside. • UNUSUAL focussing properties. • no reflection from surfaces • aberration-free focus • free from wavelength restriction on resolution 1 1 n 1 Cockcroft Institute, Spring term, 16/04/12 Coupling of unit cells •Effective continuous behaviour for wavelengths much larger than the periodicity of the inclusions (λ > 10*a) • For shorter wavelengths? (No longer effective medium but Photonic bandgap effect) • Unlimited possibilities for the design of inclusions (unit cells) Induced current Induced current in the neighbouring cell conventional material Magnetic field from incoming light Cockcroft Institute, Spring term, 16/04/12 Induced magnetic field metamaterial A bit of theory Governing equations describing EM behaviour of materials Br , t Maxwell’s equations in time domain E r , t t Dr , t J r , t H r , t t Dr , t r , t Br , t 0 Medium response description: Dr , t 0 E r , t Pr , t Br , t 0 H r , t M r , t t Pr , t 0 Rr , t t E r , t dt Frequency domain makes it easier: Constitutive relations Cockcroft Institute, Spring term, 16/04/12 Pr , 0 e r , E r , M r , m r , H r , Resonances in permittivity and permeability Lorentz model for dielectrics P 0 f E 0 E 2 2 e j •Where f is a phenomenological strength of the resonance and γ is the damping factor •Amplitude is depending on the detuning of the frequency from the resonance. •The lower the damping and the sharper the resonance, the resonance peak will also be higher and the transfer of energy from the illuminating wave to the dipole will be strongest directly at the resonance Relative permittivity: 1 Cockcroft Institute, Spring term, 16/04/12 Resonances in permittivity and permeability Drude model for metals A special case of Lorentz model: 0 p2 2 j At low frequencies the response is dominated by the imaginary part Cockcroft Institute, Spring term, 16/04/12 j Equivalent circuits of MTMs Example: permeability of artificial magnetic structures: the response is due to resonant oscillating currents. Equivalent circuit equivalent RLC circuit and apply the circuit theory equations H H 0 e j t U L U C U R U ind 1 Idt IR C U lt I I I R ind 2 0 b H 0e jt L LC L L LI I0 0lt b / L H0 2 1 LC j R L 2 / 0 1 m M N LC V lt b I M 0 m H 1 VUC 0 lt b 2 L 2 0 1 2 2 j LC Cockcroft Institute, Spring term, 16/04/12 LC 1 LC R L Designing the EM response • MTM properties are mainly due to the cellular architecture and also depend on the PCB substrate. • Great flexibility to control the EM propagation through MTMs • Material properties are characterised by an electric permittivity ε and a magnetic permeability µ. Cockcroft Institute, Spring term, 16/04/12 EM modelling of MTMs In order to verify the interesting MTM functionalities we need to retrieve the effective parameters in order to know how MTM affects light propagation. Exact material configuration (geometry + parameters of all constituents) E, H field distribution in every unit cell with periodic excitation MODEL Cockcroft Institute, Spring term, 16/04/12 S-parameters (Reflection and Transmission coefficients) Retrivial method S-parameters are defined in terms of reflection coefficient R and transmission coefficient T: S11 R S 21 Te jk0 d Z n 1 S11 2 S212 1 S11 2 S212 e jnk0 d 1 Im ln e jnk0 d 2m j Re ln e jnk0 d k0 d Cockcroft Institute, Spring term, 16/04/12 S 21 Z 1 1 S11 Z 1 A numerical example Negative index MTM (unit cell): SRR for magnetic resonance and wire for electric resonance (copper on FR4 substrate) S-parameters computed by CST Microwave Studio Cockcroft Institute, Spring term, 16/04/12 A numerical example Calculated permittivity and permeability from S-parameters: Smith, Physical Review E, 71, 036617, (2005) Cockcroft Institute, Spring term, 16/04/12 Characterisation Which post-processing can be applied to obtain effective material parameters? EM characterisation of different types of MTM is still a challenge. Most known measurement techniques for linear EM characterisation of nano-structured layers and films: Techniques Measurement equipment Direct results Spectroscopy (optical range) Precision spectrometer Absolute values of S-parameters of a layer (film) Interferometry (optical range) Precision interferometer Phase of S-parameters of a layer (film) THz time domain spectrometry (optical range) Detection of the phase change of THz wave passing through the MTM sample, compared to reference Real and imaginary part of effective refractive index via phase measurements Free space techniques (RF range) Receiving antenna + vector network analyser Complex S-parameters of the all set up between input and output ports Waveguide techniques (RF range) Receiving probe + vector network analyser Complex S-parameters of the all set up between input and output ports Cockcroft Institute, Spring term, 16/04/12