Photonic Crystals Seminar. pho·ton The quantum of electromagnetic energy, regarded as a discrete...

Photonic Crystals

Seminar

pho·ton

The quantum of electromagnetic energy, regarded as a discrete particle having zero mass, no electric charge, and an indefinitely long lifetime.

crys·tal

A homogeneous solid formed by repeating, three-dimentional pattern of atoms, ions, or molecules and having fixed distances between constituent parts. (from Greek krustallos)

The American Heritage® Dictionary of the English Language

Photonic Crystals

Example - Optical CommunicationsIn current communications systems, audio/video signals are encoded as streams of digital data packs. These voltage pulses are converted into short pulses of light that are sent along an optical-fibre network using TDM. In order to increase the amount of data that can be transmitted is by making the incoming electronic pulses as short as possible, or in other words, we need bright LEDs that can be switched on and off at very high speed.

Another way to increase the capacity of the fibre is by adding new signals at different wavelengths, a method known as DWDM. However the optical fibre is only transparent over a small range of wavelengths, so, the number of seperate transmisions depends on the linewidth of the neightbouring optical channels. Although LEDs offer high switching speeds they emit light over a wide range of wavelengths, which make them less suitable.

LEDs emit photons in many different directions, we can create LED that only emits light in the forward direction by placing a reflector behind the photoemissive layer, but the efficiency of such device is limited t the efficiency of the reflector.

By using photonic crystals one can create: 1) a novel LEDs that emit light in a very narrow wavelength range 2) a highly selective optical filter. 3) A mirror that refelcts a selected wavelength of light from any angle with high efficiency. they can also be integrated within the photoemissive layer and to create an LED that emits light at a specific wavelength and direction.

TDMTime-Division Multiplexing

a type of multiplexing that combines data streams by assigning each stream a different time slot in a set. TDM repeatedly transmits a fixed sequence of time slots over a single transmission channel. Within T-Carrier systems, such as T-1 and T-3, TDM combines Pulse Code Modulated (PCM) streams created for each conversation or data stream.





DWDMDense Wavelength Division Multiplexing

DWDM works by combining and transmitting multiple signals simultaneously at different wavelengths on the same fiber. In effect, one fiber is transformed into multiple virtual fibers. So, if you were to multiplex eight signals into one fiber, you would increase the carrying capacity of that fiber from 2.5 Gb/s to 20 Gb/s. Currently, because of DWDM, single fibers have been able to transmit data at speeds up to 400Gb/s. And, as vendors add more channels to each fiber, terabit capacity is on its way. A key advantage to DWDM is that it's protocol and bit-rate independent. DWDM-based networks can transmit data in IP, ATM, SONET /SDH, and Ethernet, and handle bit-rates between 100 Mb/s and 2.5 Gb/s. Therefore, DWDM-based networks can carry different types of traffic at different speeds over an optical channel.





What is a Photonic Crystal?

P. C. == Artificial crystal structure that could manipulate beams of light in the same way that silicon and other semiconductors control electric current.Photonic bandgap a range of forbidden frequencies within which a specific wavelength is blocked, and light is reflected. By designing the spectral and spatial location of the gap, one can almost mold the flow of light at will.

Photonic Crystals Major TimeSteps

1987

1991

1992

1994

1998

1999

Bell Communication Research, New Jersy, Eli Yablonovitch proposed the photonic-crystals structures

Yablonovitch produced the first photonic crystal by mechanicallydrilling holes in a material with n=3.6

Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ., manufactured a photonic crystal in a layer-by-layer fashion

Judith Wijnhoven and Willem Vos, Amsterdam Univ., created an almost truly 3D photonic band-gap by using sub-micron-sized silica spheres.

S.Y. Lin, Sandia National laboratories, N.M., designed a 3d photonic crystal operating at infrared wavelengths

Philip St.J. Russell, University of Bath, England, demonstrated photonic band-gap fibers.

Yablonovite1991, Bell Communications Research, N.J., Yablonovitch and co-workers suggested artifical three dimentional periodic sturctures to manipulate the propagation of light. The structure suggested, was built on a lengthscale of millimeters and shown to have a complete photonic band gap, that is to prohibit the propagation of microwaves in all directions for certain wavelengths.'Yablonovite' is made by covering a slab of material with a mask consisting of a triangular array of holes. Then, each hole is driled through three times at an angle of 35.26 from the normal and spread out 120 degrees on the azimuth. This forms a network of intersecting holes below the surface with an FCC structure.


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1991

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mm-wave photonic band-gap crystals

Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ., assembled by hand photonic crystal the size of a ping-pong balls, using the common straight metal pins used by tailors. These Photonic crystals made for microwave wavelenghts.

Later, the two fabricated a photonic crystal by stacking micromachined Si wafers in a "woodpile" or "picket fence" structure, however they had extreme difficulties in achieving the required accuracy as the dimensions of the structure were reduced and also, as the number of layers increased, in order to make a device that operates at optical wavelengths.


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Photonic Crystals made of Air SpheresJudith Wijnhoven and Willem Vos, University of Amsterdam. The two created a latttice by using sub-micron-sized silica spheres, filled the air voids between the spheres with a titania dioxide (TiO2) solution, The solution undergoes a chemical reaction that causes solid titania to form, After several times the silica spheres are dissolved leaving a close-packed lattice of spherical air voids in a titania matrix.


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3D I-R Photonic CrystalA three-dimentional photonic crystal was created using layer-by-layer periodic structure method. It is consisted by alumina rods with a stacking sequence that repeats itself every four layers with repeat distance of c. Within each layer, the axes of the rods are parallel to each other with a pitch of d. The orientations of the axes are rotated by 90o between adjacent layers. Between every other layer the rods are shifted relative to each other by 0.5d. The result is a face-centered-tetragonal (f.c.t) lattice symmetry. A computed photonic density of states plot summed over the entire brillouin zone for this lattice was computed


1987

1991

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1999







Photonic band-gap fibers

1999, Russell of the university of Bath, England, showed that like in an optical fiber, light can travel along a central hole in the fiber, confined by the two-dimensional band gap of the surrounding material. More optical power can be sent through such a central void than through glass, by so, enabling greater information carrying capacity.

Basics of photonic band gaps

The behaviour of light in a photonic crystal is similar to the movement of electrons and holes in a semiconductor.

If the periodicity of the lattice is broken by a missing Si atom or by various impurity an electron can have enough energy to be in the band gap.The same for photons in a photonic lattice, photons move in a transparent dielectric material that contains tiny air holes arranged in a lattice pattern.

We have Maxwell’s equations:(1.1)

(1.2)

(1.3)

(1.4) Where the displacement vector D is:(1.5)And,(1.6)Equation (1.5) and (1.6) are known as the Constitutive RelationsWe’re discussing the simplest case, Isotropic, linear and homogeneous material hence, ε is a scalar, there is no current (J=0) and no free charges (ρ=0), assuming non magnetic material (M=0) and we have:(1.7)(1.8)Equation (1.1) is now:(1.9)

BE

t

0 0 0

EB J

t

E 0B

0D=ε E+P

0B H M

D E

0B H

DH J

t

So the Maxwell equations have the form:

EH

t

B

Et

0E

(a) (c)

(b) (d) 0H

Substituting equation (1.8) in equations (1.1) and (1.2), getting(1.11)

(1.12) These equation were received under the assumption that the material does not change its properties because of the wave,

Executing curl on equation 1.11 getting,(1.13)

Substituting equation 1.10c,(1.14)

0 0B H

E Et t

0E

Ht

t

0( ) 0H

Et

2

0 2( ) 0

EE

t

According to vector rules, (1.15)

Equation (1.14) is now,(1.16)

The same procedure for the H field, resulting in(1.17)

The solutions for equation (1.14) and (1.15) are:(1.18)

(1.19)

Substituting back in the Maxwell’s equations,(1.20)

(1.21)

Dividing (1.20) by ε and performing curl,(1.22)

2 2( )E E E E

22

0 2

EE

t

22

0 2

HH

t

( )

0( , ) .i t k rE r t E e C C

( )

0( , ) .i t k rH r t H e C C

( ) ( ) 0i

E r H rc

( ) ( ) 0H r i E rc

1( ) ( )H r i E r

c

Using equation (1.20),

(1.23)

This equation is referred to as the MASTER equation, with the divergence equation it completely determines , hence, for a given photonic crystal with a known , one must solve the master equation to find the modes for a given frequency, then, by using (1.19) again to find E(r) :

(1.24)

21

( ) ( )H r H rc

)(rH )(r)(rH

( ) ( )c

E r i H r

1D Photonic Lattice

The material is periodic in the z-direction, and homogeneous in the xy-plane.The solution to the master equation can be written as:

because of periodicity, u(z)=u(z+R) where R=na, also, for simplicity, we restrict kz to the 1st Brillouin Zone. If the primitive lattice vector is then the primitive reciprocal lattice vector is and the Brillouin zone is

Considering light that propagate only in the z-direction through 3 different1d lattice samples,

1) a completely homogeneous medium, in this case we know that the speed of light is reduced by the index of refraction, so the frequency spectrum is given by . Because k repeates itself outsidethe Brillouin zone, the lines fold back into the zone when they reach the edges.

1) a completely homogeneous medium, in this case we know that the speed of light is reduced by the index of refraction, so the frequency spectrum is given by . Because k repeates itself outsidethe Brillouin zone, the lines fold back into the zone when they reach the edges.

2) a structure with alternating dielectric constants ε =13 and ε=12 this is a GaAs / GaAlAs multilayer. We see a gap in the frequency between the upper and lower branches of the lines-a photonic band gap. This gap occurs at the edge of the Brillouin zone at k=π/a

3) a structure with layers alternating ε=13 and ε =1 -- that is a GaAs / Air multilayer. In this structure the band gap has increased considerably

2d Photonic Lattice

The material in question for this lattice, has periodicity in 2 dimensions (that's where the photonic gaps appear) and is homogeneous in the third dimension.

Just as before, we solve the equation where now, kx and ky are restricted to the brillouin zone, while kz is unrestricted. As before, u(ρ)=u(ρ +R) where R is a lattice vector, but now, u is periodic in the xy-plane and not in the z-direction as for the 1d lattice.

Assuming kz=0, that is, light is propagating only in the xy-plane we'll be looking at the TE (Transverse Electric) and TM (Transverse Magnetic) modes, we are referring to three special points Γ, X and M that correspond to kx,y=0, kx,y=π/a x;

and kx,y= π/a x + π/a y. the rest of the brillouin zone can be related to this triangle by rotational symmetry.

Assuming kz=0, that is, light is propagating only in the xy-plane we'll be looking at the TE (Transverse Electric) and TM (Transverse Magnetic) modes, we are referring to three special points Γ, X and M that correspond to kx,y=0, kx,y=π/a x;

and kx,y= π/a x + π/a y. the rest of the brillouin zone can be related to this triangle by rotational symmetry.

In this arrangement, holes of ε=1 (air) in a dielectric material ε=8.3 (alumina) we get a different result of the band gap.

Note: TM band gaps are favored in a lattice of isolated high-ε regions, and TE band gaps are favored in a connected lattice

Bandgap Calculation for 2d Photonic Crystal

For simplicity we will be using this lattice:

Because of periodicity of the lattice it is possible to write

Where R is one the original lattice vectorsUsing Bloch’s theorem we write:

And because of the periodicity it is possible to prove that:

The solution of this equation is found when:

which can be interpreted to:n Є N

)()( Rrr

rkiknnk erurEE

)()(

)()( Rruru nknk

rkinknk erERrE

)()(

1Rkie

nRk 2

Our lattice vectors can be defined

where en are the lattice vectors and ηn are integers, for 2D η3=0. Similarly,the reciprocal lattice:

where fn can be found with:

remembering that for 2D f3=0. For a spaced lattice we have: |e1|=|e2|=a and our reciprocal lattice has the dimensions:

|f1|=|f2|=4π

a·sqrt(3)

332211 eeeR

332211 fffG

)(2

321

321 eee

eef

)(2

132

132 eee

eef

we see that we get periodicity also in the reciprocal space, so the modes with the wave vector k are identical to the wave vectors k+G

For this case we can easily obtain for the three corners Γ M and K:Γ: kx=0 ky=0 M: kx=0 ky=

K: kx= ky=

The calculations are now made using variouscomputer software obtaining:

a3

23

2

a

3

2

a


K: kx= ky=


a3

23

2

a

3

2

a


K: kx= ky=


a3

23

2

a

3

2

a

Computational Physics

Although complex structures and advanced techniques are being used, fundamental questions about the formation, operation and efficiency of photonic band gaps remain unanswered.

In order to exploit Photonic Crystals to their full potential it is vital that they can be modeled accurately and efficiently. Traditionally, plane wave expansion has been used as the numerical method; this approach has produced successful results but has disadvantages such as slow convergence and high demands for computing power and memory. Finite Element Analysis

address these problems in the computational modeling process .

To be able to create photonic crystals for optical devices, we need to use state-of-the-art semiconductor microfabrication techniques with their associated high production costs and investment. For this reason, computer modelling of prospective photonic-crystal structure is also a very important area of research, as it may prevent expensive fabrication errors later.

Optical Circuits

In the following figure we see a wide Y-shaped "defect waveguide" within the photonic crystal by removing some of the holes from the lattice.

Optical FibersThe light is guided along the low-refractive-index air core by a photonic-band-gap confinement effect.

The FutureThe future for photonic crystal circuits and devices looks certain.


Within 5 yearsBasic applications, highly efficient photonic-crystal lasers;Routing light round micron-sized optical benches; Hi-res spectral filtering

5-10 years The first photonic crystal "diodes" and "transistors"






10-15 years Photonic Crystal logic circuits




10-15 years Photonic Crystal logic circuits

15-25 years Optical Computer driven by photonic crystals

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Photonic Crystals Seminar. pho·ton The quantum of electromagnetic energy, regarded as a discrete...

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