1. Photonic Crystals and Metamaterials and accelerator applications
Rosa Letizia
Lancaster University/ Cockcroft Institute
Cockcroft Institute, Spring term, 16/04/12

2. 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

3. 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

4. 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

5. 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) ]

6. 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.

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. PhC and defects
Line defect
Waveguide

15. 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)

16. 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) ]

17. 3D PhC – Woodpile structure
Unit cell: 4 layers

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

19. 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 = nm
a2 = 280 nm
r 2= nm
w = 0.38 m
a = 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 m2/W
sat = 0.31
D Pinto, and S S A Obayya, IET Optoelectron., vol. 2, no. 6, pp. 254261, 2008

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

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

22. 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

23. 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

24. 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, , 2006.
[2] A. Greenleaf, et al., Phys. Rev. Lett., 102, (1-4), 2007.
[3] X. Zhang, et al., Opt. Express, 16, , 2008.
[4] R. Liu, et al., Science, 323, , 2009.

25. 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

26. Negative refractive index
MTMs obey Snell’s law:

27. 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

28. 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

29. 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

30. 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:

31. 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

32. 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.

33. 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.

34. 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)

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

36. 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

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

38. 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