Transcript What is a Photonic Crystal? - Computational Physics/HOME

Photonic Crystals
Seminar
Photonic Crystals
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
Example - Optical Communications
In 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.
TDM
Time-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.
Example - Optical Communications
In 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.
DWDM
Dense 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.
Example - Optical Communications
In 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.
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
Bell Communication Research, New Jersy, Eli Yablonovitch
proposed the photonic-crystals structures
1991
Yablonovitch produced the first photonic crystal by mechanically
drilling holes in a material with n=3.6
1992
Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ.,
manufactured a photonic crystal in a layer-by-layer fashion
1994
Judith Wijnhoven and Willem Vos, Amsterdam Univ., created
an almost truly 3D photonic band-gap by using sub-micron
-sized silica spheres.
1998
S.Y. Lin, Sandia National laboratories, N.M., designed a 3d
photonic crystal operating at infrared wavelengths
1999
Philip St.J. Russell, University of Bath, England, demonstrated photonic band
-gap fibers.
Yablonovite
1991, 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.
Photonic Crystals Major TimeSteps
1987
Bell Communication Research, New Jersy, Eli Yablonovitch
proposed the photonic-crystals structures
1991
Yablonovitch produced the first photonic crystal by mechanically
drilling holes in a material with n=3.6
1992
Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ.,
manufactured a photonic crystal in a layer-by-layer fashion
1994
Judith Wijnhoven and Willem Vos, Amsterdam Univ., created
an almost truly 3D photonic band-gap by using sub-micron
-sized silica spheres.
1998
S.Y. Lin, Sandia National laboratories, N.M., designed a 3d
photonic crystal operating at infrared wavelengths
1999
Philip St.J. Russell, University of Bath, England, demonstrated photonic band
-gap fibers.
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.
Photonic Crystals Major TimeSteps
1987
Bell Communication Research, New Jersy, Eli Yablonovitch
proposed the photonic-crystals structures
1991
Yablonovitch produced the first photonic crystal by mechanically
drilling holes in a material with n=3.6
1992
Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ.,
manufactured a photonic crystal in a layer-by-layer fashion
1994
Judith Wijnhoven and Willem Vos, Amsterdam Univ., created
an almost truly 3D photonic band-gap by using sub-micron
-sized silica spheres.
1998
S.Y. Lin, Sandia National laboratories, N.M., designed a 3d
photonic crystal operating at infrared wavelengths
1999
Philip St.J. Russell, University of Bath, England, demonstrated photonic band
-gap fibers.
Photonic Crystals made of Air Spheres
Judith 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.
Photonic Crystals Major TimeSteps
1987
Bell Communication Research, New Jersy, Eli Yablonovitch
proposed the photonic-crystals structures
1991
Yablonovitch produced the first photonic crystal by mechanically
drilling holes in a material with n=3.6
1992
Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ.,
manufactured a photonic crystal in a layer-by-layer fashion
1994
Judith Wijnhoven and Willem Vos, Amsterdam Univ., created
an almost truly 3D photonic band-gap by using sub-micron
-sized silica spheres.
1998
S.Y. Lin, Sandia National laboratories, N.M., designed a 3d
photonic crystal operating at infrared wavelengths
1999
Philip St.J. Russell, University of Bath, England, demonstrated photonic band
-gap fibers.
3D I-R Photonic Crystal
A 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-centeredtetragonal (f.c.t) lattice symmetry.
A computed photonic density of states plot summed over the entire brillouin
zone for this lattice was computed
Photonic Crystals Major TimeSteps
1987
Bell Communication Research, New Jersy, Eli Yablonovitch
proposed the photonic-crystals structures
1991
Yablonovitch produced the first photonic crystal by mechanically
drilling holes in a material with n=3.6
1992
Ekmel Ozbay and Gary Tuttle, Ames Lab - Iowa State Univ.,
manufactured a photonic crystal in a layer-by-layer fashion
1994
Judith Wijnhoven and Willem Vos, Amsterdam Univ., created
an almost truly 3D photonic band-gap by using sub-micron
-sized silica spheres.
1998
S.Y. Lin, Sandia National laboratories, N.M., designed a 3d
photonic crystal operating at infrared wavelengths
1999
Philip St.J. Russell, University of Bath, England, demonstrated photonic band
-gap fibers.
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 twodimensional 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:
E
(1.1)
 B  0 0  0 J
(1.2)
(1.3)
(1.4)
t
B
 E  
t
 B  0
E  
Where the displacement vector D is:
(1.5)
D=ε 0 E+P
And,
B  0  H  M 
(1.6)
Equation (1.5) and (1.6) are known as the Constitutive Relations
We’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:
D E
(1.7)
(1.8)
B  0 H
Equation (1.1) is now:
(1.9)
D
 H  J 
t
So the Maxwell equations have the form:
(a)
 E  
B
t
(c)
(b)  E  0
(d)
 H  
E
t
 H  0
Substituting equation (1.8) in equations (1.1) and (1.2), getting
(1.11)   E  B    E   H  0
t
(1.12)
 H 
0
t
E
0
t
These equation were received under the assumption that the material does not change
its properties because of the wave,


t
Executing curl on equation 1.11 getting,
H
(1.13)
 ( E)     0
t
0
Substituting equation 1.10c,
E
  (  E )    
0
(1.14)
2
0
t 2
According to vector rules,
    E  (  E )   E   E
(1.15)
2
Equation (1.14) is now,
(1.16)
2
2 E
 E   0 2
t
2
The same procedure for the H field, resulting in
H
(1.17)
 H 
2
2
t 2
0
The solutions for equation (1.14) and (1.15) are:
(1.18)
E (r , t )  E0 e  i (t k r )  C.C
(1.19)
H (r , t )  H 0 e  i (t  k r )  C .C
Substituting back in the Maxwell’s equations,

(1.20)
 H (r )  i
E (r )  0
c
(1.21)
 E (r ) 
i
H (r )  0
c
Dividing (1.20) by ε and performing curl,
1

 

(1.22)
    H (r )    i E (r )





 c


Using equation (1.20),
1
  
     H (r )     H (r )

 c
2
(1.23)
This equation is referred to as the MASTER equation, with the divergence equation it
completely determines H (r ), hence, for a given photonic crystal with a known ,  (r )
one must solve the master equation to find the modes H (r ) for a given frequency, then,
by using (1.19) again to find E(r) :
(1.24)
 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 different
1d 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 outside
the 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 outside
the 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 xyplane 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  (r )   (r  R)
Where R is one the original lattice vectors
Using Bloch’s theorem we write:

  

i
k
E  Ekn (r )  ukn (r )e r
And because of the periodicity it is possible to prove
that:






ukn (r )  ukn (r  R)
  
  ik r

 Ek n (r  R)  Ekn (r )e
The solution of this equation is found when:
e
 
ik R
1
which can be interpreted to:
k  R  2n
nЄN
Our lattice vectors can be defined




R  1e1  2e2  3e3
where en are the lattice vectors and ηn are integers, for 2D η3=0. Similarly,
the reciprocal lattice:




G  1 f1   2 f 2   3 f3
where fn can be found with:
 
 


e3  e1
e2  e3
f

2

f1  2   
  
2
e2  (e3  e1 )
e1  (e2  e3 )
remembering that for 2D f3=0. For a spaced lattice we have:
|e1|=|e2|=a
and our reciprocal lattice has the dimensions:
4π
|f1|=|f2|= a·sqrt(3)
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:
2
Γ: kx=0 ky=0
M: kx=0 ky=
a 3
2
2
K: kx=
ky=
a 3
3a
The calculations are now made using various
computer software obtaining:
For this case we can easily obtain for the three
corners Γ M and K:
2
Γ: kx=0 ky=0
M: kx=0 ky=
a 3
2
2
K: kx=
ky=
a 3
3a
The calculations are now made using various
computer software obtaining:
For this case we can easily obtain for the three
corners Γ M and K:
2
Γ: kx=0 ky=0
M: kx=0 ky=
a 3
2
2
K: kx=
ky=
a 3
3a
The calculations are now made using various
computer software obtaining:
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 Fibers
The light is guided along the low-refractive-index air core by a photonic-band-gap
confinement effect.
The Future
The future for photonic crystal circuits and devices looks certain.
The Future
The future for photonic crystal circuits and devices looks certain.
Basic applications, highly efficient photonic-crystal lasers;
Within 5 years Routing light round micron-sized optical benches;
Hi-res spectral filtering
The Future
The future for photonic crystal circuits and devices looks certain.
Basic applications, highly efficient photonic-crystal lasers;
Within 5 years Routing light round micron-sized optical benches;
Hi-res spectral filtering
5-10 years
The first photonic crystal "diodes" and "transistors"
The Future
The future for photonic crystal circuits and devices looks certain.
Basic applications, highly efficient photonic-crystal lasers;
Within 5 years Routing light round micron-sized optical benches;
Hi-res spectral filtering
5-10 years
10-15 years
The first photonic crystal "diodes" and "transistors"
Photonic Crystal logic circuits
The Future
The future for photonic crystal circuits and devices looks certain.
Basic applications, highly efficient photonic-crystal lasers;
Within 5 years 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
15-25 years
Optical Computer driven by photonic crystals