2. Fundamentals of Crystals
Material properties are determined by the properties of their sub-units with
their spatial distribution.
Electromagnetic properties as a function of the ratio a the “lattice
constant” of the material/structure and λ the wavelength of the incoming
light (a/ ) can be organized in three large groups:
Natural crystals and metamaterials have lattice constants much smaller
than the light wavelengths: a << λ. These materials are treated as
homogeneous media with parameters ε and μ.
3. Fundamentals of Crystals
When a is in the same range of the wavelength of the incoming light one
defines a photonic crystal; a material with subunits bigger than atoms but
smaller than the EM wavelength.
In photonic crystals a is the distance between repeat units with a different
dielectric constant.
4. Photonics Crystals
Photonic crystals are periodic optical nanostructures that affect the
motion of photons in much the same way
that semiconductors affect electrons.
Photonic crystals have properties governed by the diffraction of the
periodic structures and may exhibit a band gap for photons.
Photons (behaving as waves) propagate through this structure – or not –
depending on their wavelength.
Wavelengths that are allowed to travel are known as modes; groups of
allowed modes form bands.
Disallowed bands of wavelengths are called photonic band gaps
5. Photonics Crystals
They typically are not described well using effective parameters ε and μ
and may be artificial or natural.
In 1987 Sajeev John and Eli Yablonovitch proposed of photonics crystals
with periodicity of n in 2D and 3D.
1D crystals (example Braggs Mirror or Distributed Bragg Reflector) were
known since 1887 .
6. Distributed Bragg Reflector
formed from multiple layers of alternating materials with varying refractive
index with each layer boundary causes a partial reflection of an optical
wave.
for waves whose wavelength is close to four times the optical thickness of
the layers, the many reflections combine with constructive interference,
and the layers act as a high-quality reflector.
The range of wavelengths that are reflected is called the photonic stop
band . Within this range of wavelengths, light is "forbidden" to propagate in
the structure.
12. Bragg’s Scattering
Regardless of how small the reflectivity r form an individual scatter, the
total reflectivity R for a semi-infinite structure is given by :
13. Photonic Band Gap
So light cannot propagate in a crystal when frequency of incident light
satisfies Bragg’s Condition :
Photonic Band Gap (PBG)
14. Photonic Band Gap
In a periodic system, when half the wavelength corresponds to the
periodicity i.e., λ/2 = a then Bragg’s Condition K= π/a prohibits photon
propagation
15. Band Structure of 1D Photonics Crystal
The dispersion curve of a 1D “photonic crystal” deviates from the straight-
line dispersion curve of a uniform bulk medium.
16. Band Structure of 1D Photonics Crystal
This is because at k=π/a formation of standing waves occur which have
zero group velocity discontinuity at that point
The energy of Standing waves being either in the high or the low index
regions therefore we have dielectric band & air band
21. Photonics in Nature
In Parides sesostris, the Emerald-patched Cattleheart butterfly, photonic
crystals are formed of arrays of Nano - sized holes in the chitin of the wing
scales.
The holes have a diameter of about 150 nanometers and are about the same
distance apart.
The holes are arranged regularly in small patches; neighboring patches
contain arrays with differing orientations.
The result is that these Emerald-patched Cattleheart scales reflect green light
evenly at different angles instead of being iridescent.
Iridescence is generally known as the property of certain surfaces that appear
to change color as the angle of view or the angle of illumination changes
23. Photonics Crystal Application
Most proposals for devices that make use of photonic crystals do not use
the properties of the crystal directly but make use of defect modes.
Such a defect is made when the lattice is changed locally. As a result, light
with a frequency inside the bandgap can now propagate locally in the
crystal, i.e. at the position of the defect.
24. Optical Fiber
An optical fiber is a cylindrical dielectric waveguide (non conducting
waveguide) that transmits light along its axis, by the process of total
internal reflection.
The fiber consists of a core surrounded by a cladding layer, both of which
are made of dielectric materials.
To confine the optical signal in the core, the refractive index of the core
must be greater than that of the cladding.
Light travels through the fiber core, bouncing back and forth off the
boundary between the core and cladding.
25. Photonic Crystal Fiber
Photonic crystal optic fibers are a special class of 2D photonic crystals
obtains its waveguide properties not from a spatially varying glass
composition but from an arrangement of very tiny and closely spaced air
holes which go through the whole length of fiber.
the simplest type of photonic crystal fiber has a triangular pattern of air
holes, with one hole missing i.e. with a solid core surrounded by an array of
air holes.
26. Photonic Crystal Fiber
The guiding properties of this type of PCF can be roughly understood with
an effective index model: the region with the missing hole has a higher
effective refractive index, similar to the core in a conventional fiber.
The gray area indicates glass, and the white circles air holes with typical
dimensions of a few micrometers.
27. Photonic Band Gap Fibers
based on a photonic bandgap of the cladding region
The refractive index of the core itself can be lower than that of the
cladding structure.
Essentially, a kind of two-dimensional Bragg mirror is employed.
28. Metamaterial Photonics
In photonic crystals, the size and periodicity of the scattering elements are
on the order of the wavelength rather than subwavelength.
subwavelength is used to describe an object having one or more
dimensions smaller than the length of the wave with which the object
interacts.
At optical frequencies(of GHz order) electromagnetic waves interact with
an ordinary optical material (e.g., glass) via the electronic polarizability of
the material.
This creates a state where the effective permeability of the material is
unity, μeff = 1
29. Metamaterial Photonics
Hence, the magnetic component of a radiated electromagnetic field has
virtually no effect on natural occurring materials at optical frequencies.
However, the proper design of the elementary building blocks of the
photonic metamaterial allows for a non-vanishing magnetic response and
even for μ<0 at optical frequencies.
Photonic metamaterials, are a type of electromagnetic
metamaterial, which are designed to interact with optical frequencies
which are terahertz (THz), infrared (IR), and eventually, visible wavelengths.
30. Structures Containing Nano-Resonators
Photonic metamaterials typically contain some kind of metallic
nanoscopic electromagnetic resonators.
An early approach, which has been taken over from previous work in the
microwave domain, is based on split-ring resonators.
The resonances of such a resonator can be in the mid-infrared domain
(with wavelengths of a few microns) when its width is reduced to the order
of a few hundred nanometers.
A magnetic field, oriented perpendicular to the plane of the rings, induces
an opposing magnetic field due to the Lenz’s law, which leads to a
diamagnetic response resulting in a negative permittivity in a certain
range of frequencies
32. Metamaterial Photonics
When light impinges such nano-resonators, it can excite electromagnetic
oscillations.
These are particularly strong for frequencies near the resonance
frequency.
As the period of the structure is well below half the optical wavelength,
there are no photonic bandgap effects, and the effect on light
propagation can be described with a (frequency-dependent) effective
relative permittivity ε and relative permeability μ of the metamaterial
33. Metamaterial Photonics
The electric resonances of individual nanorods originate from the
excitation of the surface waves on the metal air interface.
In a paired nanorod configuration two types of plasmon polariton waves
can be supported: symmetric and anti-symmetric.
34. Problems Encountered
Constructing Photonics Materials in near-infrared and visible frequencies
turned out not to be straightforward for at least two reasons:
1. technical challenges related to the fabrication of resonant structures on
the nanoscale .
2. resonance frequency saturates as the size of the SRR reduces, and the
amplitude of the resonant permeability decreases
Modern nanofabrication techniques such as Scanning Electron Beam
Lithography enable the fabrication of optical components on the scale of
the optical wavelength with a relative precision in the few nanometer
range
35. References
E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
R. A. Depine and A. Lakhtakia (2004). "A new condition to identify isotropic
dielectric-magnetic materials displaying negative phase velocity".
Microwave and Optical Technology Letters 41.
Veselago, V. G. (1968). "The electrodynamics of substances with
simultaneously negative values of [permittivity] and [permeability]". Soviet
Physics Uspekhi 10 (4): 509–514.
36. References
S. John, Phys. Rev. Lett. 58, 2486 (1987).
Advances in Complex Artificial Electromagnetic Media by Nathan Kundtz
Department of Physics , Duke University.
K. Ohtaka, Phys. Rev. B 19, 5857 (1979)
Schurig,, D. et al. (2006). "Metamaterial Electromagnetic Cloak at
Microwave Frequencies".
