Photonic Crystals and Metamaterials and accelerator applications Rosa Letizia Lancaster University/ Cockcroft Institute r.letizia@lancaster.ac.uk 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

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

What are Photonic crystals (PhCs)? 1-D PhC 2-D 3-D 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 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 3µm [ P. Vukosic et al., Proc. Roy. Soc: Bio. Sci. 266, 1403 (1999) ] [ 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. Constructive interference when the difference of optical path is equal to an integer number of wavelengths. Diffraction and scattering at the dielectric interfaces can build in a destructive way reflected and incident waves so that light is forbidden to propagate. This process results in the creation of photonic bandgaps.

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) Plane wave Has the periodicity of the crystal lattice BLOCH MODES

Maxwell’s equations in periodic media 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 The mode which corresponds to k is equal to the one at k+G that corresponds to an increment of G*R in the phase equal to an integer multiple of 2pi. Thus it is possible to constrain the study of k to a finite range of values in which non redundant modes are obtained. This range is called BRILLOUIN ZONE Brillouin Zone

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 Let’s consider a 1D uniform dielectric with artificial periodicity a and plane waves ck as solutions. a is artificial periodicity 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 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

2D Photonic Crystals Rectangular lattice Hexagonal lattice The frequencies are expressed in terms of c/a. Thanks to translational symmetry, Maxwell’s equations solutions are invariant under translations of distances that are multiples of some fixed step lengths (period a) In a 2D system the EM fields can be divided into 2 polarisations depending on their symmetry: TE and TM A pbg requires E to be concentrated in narrow regions therefore the first PhC is more suitable for TE since it has small spots of high n and the E is parallel to the pillars. The second is more suitable for TM due to the presence of dielectric interconnection region. Hexagonal lattice

2D Photonic Crystals The frequencies are expressed in terms of c/a. Thanks to translational symmetry, Maxwell’s equations solutions are invariant under translations of distances that are multiples of some fixed step lengths (period a) In a 2D system the EM fields can be divided into 2 polarisations depending on their symmetry: TE and TM A pbg requires E to be concentrated in narrow regions therefore the first PhC is more suitable for TE since it has small spots of high n and the E is parallel to the pillars. The second is more suitable for TM due to the presence of dielectric interconnection region.

Breaking the periodicity: Point defect PhC and defects Breaking the periodicity: Point defect Cavity By perturbing the periodicity and thus destroying the translational symmetry of the lattice, a localised state is introduced in the forbidden gap. It is no longer possible to classify modes into in-plane wavevectors Joannopoupos, Photonic Crystals Molding the flow of light: jdj.mit.edu/book

PhC and defects Line defect Waveguide

PhC and defects EM fields highly confined in defect regions 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)

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) [ 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

PhC-based optical filter* PhC applications PhC-based optical filter* *R. Letizia, and S. S. A. Obayya, IET Optoelectronics, vol. 2, n. 6, pp. 241-253, 2008.

PhC applications y x c = a a1 = 262 nm w = 0.38 m n1 = 1.0 Photonic wire (PhW) x y c = a n1 = 1.0 n2 = 3.48 a1 = 262 nm r1 = 51.93 nm a2 = 280 nm r 2= 63.94 nm w = 0.38 m a = 0.3162 m r = 0.23a Hibryd photonic crystal since we use index guiding (confinement) in the transversal plane (conventional optical waveguide we guide by making the core high index compared to cladding) while along the propagation direction we use the principle of photonic bandgap. 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. 254261, 2008

PhC applications Linear res = 1.536 m Q = 427 T = 0.66 Nonlinear

PhC applications Fundamental frequency (TM) Enhanced Second Harmonic Generation Fundamental frequency (TM) SH frequency (TE) SH frequency

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” Metamaterials are one of the new discoveries of the last 20 years (the field began in ’90 with the pioneer work of John Pendry). They present exceptional many are still unexploited properties dominated by their geometrical structure. The structural elements or inclusions are unit cells (small building blocks) (artificial molecules) in large 1D 2D or 3D arrays, the possibilities of combining different inclusions shapes are infinite and so are the possibilities for the artificial material behaviour

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

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]. MTM are often associated with negative refraction and this property has gained substantial attention because of its potential for clocking invisibility devices and microscopy with super-resolution. [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.

Backward-wave materials 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

Negative refractive index MTMs obey Snell’s law:

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

Coupling of unit cells Induced current 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 Magnetic field from incoming light Induced magnetic field Induced current in the neighbouring cell conventional material metamaterial

A bit of theory Governing equations describing EM behaviour of materials Maxwell’s equations in time domain Medium response description: The fact that chi is a function of frequency expresses that the medium has ‘memory’ which means polarisation density does not depend only on instantaneous value of the field amplitude but also on its past values. Frequency domain makes it easier: Constitutive relations

Resonances in permittivity and permeability Lorentz model for dielectrics 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 In dielectric materials the charges are displaced by an incident electric field. The amplitude is given by the restoring forces on the electrons and depends strongly on the time-variation of the field and this dipole formed by the displaced electron and the remaining ion will oscillate. For frequencies around a certain frequency omega_e, the induced dipole moment may be very large and strongly depend on the frequency. This exceptional interaction called “resonance” of incident EM field with the structure will lead to strong scattering and absorption. The behaviour of the oscillating dipole can be described on the base of a simple driven harmonic oscillator and the solution gives the relation between P and E (constitutive relation) expressed by susceptibility chi. The relation usually reads ... Relative permittivity:

Resonances in permittivity and permeability Drude model for metals A special case of Lorentz model: At low frequencies the response is dominated by the imaginary part Typical conductivity behaviour for metals at low frequencies

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 Using this theory for a “lattice” of magnetic elements one can obtain an expression for the current in the structure and through it one for the magnetic moment and the frequency dependency for mu(omega). R C and L will depend on the geometry of the inclusions in the unit cell. Let’s consider a system of slab pairs excited by H perpendicular to the pair plane. At the magnetic resonance of the structure resonant antiparallel currents are excited in the two slabs creating an effective loop current. This current results in accumulation of opposite charges at the two upper and two lower sides of the slabs creating capacitive regimes there. The effective circuit approximates this effect. We have L, inductance of loop current, C=C1/2, capacitance of the two capacitive regimes. Phi = external magnetic flux, Uind =voltage induced by external magnetic flux. NLC = number of LC circuits, V = volume, VUC = unit cell volume A part from the omega square at the numerator, this formula represents a Lorenzian oscillator resonance for a magnetic atom. As it is done in many cases, we can lump losses and scattering into the damping factor gamma at the denominator.

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.

MODEL 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. E, H field distribution in every unit cell with periodic excitation MODEL Exact material configuration (geometry + parameters of all constituents) S-parameters (Reflection and Transmission coefficients)

Retrivial method S-parameters are defined in terms of reflection coefficient R and transmission coefficient T: D= thickness of the material being examined

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

A numerical example Calculated permittivity and permeability from S-parameters: Smith, Physical Review E, 71, 036617, (2005)

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