通信システム工学B

Communications Devices
Lecture on 6/23, 6/27, 6/30, 7/7, 7/14, 7/21 in 2015
Hirohito YAMADA
About lecture
1. Schedule
6/23 Basic of semiconductor photonic devices
6/27 Matter-electromagnetic wave interaction based on semi-classical theory
6/30 Electromagnetic field quantization and quantum theory
7/7 Optical transition in semiconductor, Photo diode, Laser diode
7/14 Optical amplifier, Optical modulator, Optical switch,
Optical wavelength filter, and Optical multiplexer/demultiplexer
7/21 Summary
2. Textbook written in Japanese
米津宏雄著、光通信素子工学 - 発光・受光素子 -、工学図書
霜田光一編著、量子エレクトロニクス、裳華房
山田実著、電子・情報工学講座15 光通信工学、培風館
伊藤弘昌編著、フォトニクス基礎、朝倉書店第5章
3. Questions E-mail: [email protected], or ECEI 2nd Bld. Room 203
4. Lecture note dounload URL: http://www5a.biglobe.ne.jp/~babe
Data traffic explosion on the Internet
Trend of data traffic processed by a domestic network node
Data rate (Gbit/s)
Rapid increase of double per 2.5 years
Double/2.5 years
Double/5 years
double/year
Double/3 years
M/D/Y
http://www.jpix.ad.jp/en/technical/traffic.html
Growth of internet traffic in Japan
Total download traffic in Japan was about 2.6T bps at the end of 2013
Daily average value
Annual growth rate: 30%
Total download traffic in Japan
Total upload traffic in Japan
2004
2005
2006
2007
2008
2009
year
2010
2011
2012
2013
Cited from: H26年度版情報通信白書
Optical fiber submarine networks
Cited from http://www1.alcatel-lucent.com/submarine/refs/index.htm
Power consumption forecast of network equipments
Total internet traffic (Tbps)
Annual power consumption of
network equipments (×1011 Wh)
Domestic internet traffic is increasing 40%/year
If increasing trend continue, by 2024, power consumption of
ICT equipments will exceed total power generation at 2007
Total power generation
at 2007
year
http://www.aist-victories.org/jp/about/outline.html
Expanding applied area of optical communication
Nowadays, application area of optical communications are spreading
from rack-to-rack of server to universal-bass-interface of PCs
Backplane of a server
(Orange color cables are optical fibers)
Universal Bass interface (Light Peak)
installed in SONY VAIO Z
Photonic devices used for optical communications
Various optical devices for use in optical networks
1. Passive optical device, Passive photonic device
- Optical waveguide, optical fiber
- Optical splitter
: Treated in our lecture
- Optical directional coupler
- Optical wavelength filter
- Wavelength multiplexer/demultiplexer (MUX/DEMUX)
- Light polarizer
- Wave plate
- Dispersion control device
- Optical attenuator
- Optical isolator
- Optical circulator
- Optical switch, Photonic switch
- Photo detector, Photo diode (PD)
2. Active optical device, Active photonic device
- Light-emitting diode (LED)
- Semiconductor laser, Laser diode (LD)
- Optical amplifier
3. Other devices(Wavelength converter, Optical coherent receiver, etc.)
Photonic devices: supporting life of 21st century
Various photonic devices used for applications other than optical communication
- charge-coupled device (CCD) image sensor
- CMOS image sensor
- solar cell, photovoltaic cell
- photo-multiplier
- image pick-up tube
- CRT: cathode-ray tube, Braun tube
- liquid crystal display (LCD)
- plasma display
- organic light emitting display
- various recording materials (CD, DVD, BLD, hologram, film, bar-code)
- various lasers (gas laser, solid laser, liquid laser)
- non-linear optical devices
These devices collectively means “Photonic devices”
What is Photonic Device ?
Manipulating electron charge
Electronics
Magnetic Recording
voltage and current
Manipulating
Electron Tube, Diode, Tr, FET, LSI wavefunction of electron
Tunnel effect devices
Manipulating
both spin and Spintronics
charge of
GMR HDD
electrons
MRAM
Photo-electronics
(Opt-electronics)
Display
CCD, CMOS sensor Manipulating
photon and
solar cell
electron charge
LD, LED
Unexplored Photo detector, PD Manipulating
photon
Magnetics
Photonics
Opto-spinics
Energy and
number
Laser
magnet-optical
HDD
Optical disk
Electromagnetic
disk
Magnetic tape
wave
Optical fiber
Manipulating spin
amplitude and
of materials
phase
?
Basis of semiconductor photonic devices
Properties required for semiconductor used as photonic devices
105
・ Transparent at operating wavelength
・ Small nonlinear optical effect
(as distinct from nonlinear optical devices)
・ Better for small material dispersion
・ Small birefringence
(polarization independence)
Active devices(Light-emitting devices)
・ Moderately-opaque at operating
wavelength
・ High radiant transition probability
in case of light-emitting devices
(Direct transition semiconductor )
・ To be obtained pn-junction (To be
realized current injection devices)
Optical absorption coefficient: α (cm-1)
Passive devices(Non light-emitting devices)
T = 300K
Si
104
In0.53Ga0.47As
GaAs
InP
Ge
103
102
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Wavelength: λ (μm)
Optical absorption coefficient of
major semiconductor materials
Band structure of semiconductors
Semiconductor and band structure
According to Bloch theorem, wave function of an electrons in crystal
is described as a quantum number called “wave number” This
predicts existing dispersion relation between energy and wave
number of electron. This relation is called energy band (structure)
Electron
Hole
In bulk Si, holes distribute at around
the Γ point, on the other hand,
electrons distribute at around the X
point (Indirect transition
semiconductors)
Band gap ~1.1eV
Dispersion relation of electron energy in Si
Band structure of semiconductors
Band structure of compound semiconductors
Both electrons and holes distribute at around the Γ point
(Direct transition semiconductors)
Electron
Electron
Hole
Hole
GaAs
InP
Band structure of semiconductors
Band structure of Ge
Ge which is group Ⅳ semiconductor is also
indirect transition type semiconductor, but
by adding tensile strain, it changes a direct
transition-like band structure
Recent year, Ge laser diode(RT, Pulse)
was realized by current injection
Band structure of Ge
Conduction
band
Conduction
band
Indirect transition
Direct transition
1.6%
tensile strain
Valence band
Valence band
Basis of semiconductor photonic devices
Material dispersion
Values of dielectric constant (refractive index), magnetic permeability depend
on frequency of electromagnetic wave interacting with the material
Material equation D   ( ) E , B   ( ) H
Dielectric constant (refractive index) significantly
changes at the resonance frequency of materials.
In linear response, between real part and
imaginary part of frequency response function
holds Kramers-Kronig relation.
Real part
 ( ), n( ),  ( )
Photon energy (eV)
Phenomenologically-derived equation of relation
between wavelength and refractive index
Ai 2
n  1  2 2
  i
Here, λi = c/νi, c: light speed, νi: resonance
frequency of material, Ai: Constant
Imaginary part
W. Sellmeier equation
2
Calc.
Calc.
Photon energy (eV)
Calculation of dielectric function ε of Si
Basis of semiconductor photonic devices
Birefringence
In anisotropic medium, dielectric constant (refractive index), magnetic permeability
are tensor
 Dx    xx  xy  xz  E x 
 
  
 Dy     yx  yy  yz  E y 
 
 D  
 z   zx  zy  zz  E z 
(Material equation)
Incident ray
Optical axis
Outgoing ray
extraordinary
ray
Crystal is optically anisotropic medium
When light beam enter a crystal, it splits two
beams (ordinary ray and extraordinary ray)
ordinary ray
In birefringence crystals, an incident direction
where light beam does not split is called optical
axis (correspond to c-axis of the crystal)
Birefringence of calcite
Basis of semiconductor photonic devices
Nonlinear optical effect
Values of dielectric constant (refractive index), magnetic permeability depend
on amplitude of electromagnetic field interacting with the material
 ( E ), n( E ),  ( H )
Material equation D   ( E) E , B   ( H ) H
When strong electric field (light) applied to a material, nonlinear optical effects
emerge. Wavelength conversion devices use this effects.
When intensity of incident light is weak, linear polarization P which proportional
to the electric field E is induced.
D( x, t )   E ( x, t )   0 r E ( x, t )
  0 (1   e ) E ( x, t )   0 E ( x, t )   0  e E ( x, t )
  0 E ( x, t )  P ( x, t )
Linear polarization
P  0 E
 : Electric susceptibility
When intensity of incident light become strong, electric susceptibility become
depended on the electric field E
P   0  ( E) E
 (E)   0  (1)   (2) E   (3) EE   (4) EEE  
Basis of semiconductor photonic devices
Electron transition in direct transition type
and indirect transition type semiconductors
In case of indirect transition, phonon
intervenes light emission or absorption
Band structure of (a) GaAs and (b) Si
Light emission from materials
Light absorption and emission in material
Excited state
Nucleus
All light come from atoms !
Sun light‥‥Nuclear fusion of hydrogen
Fluorescence of fireflies ‥‥Chemical reaction of
organic materials
Light from burning materials ‥‥ Chemical reaction
of organic materials
Electroluminescence from LED‥‥Electron transition ΔE
in semiconductors
Light
Ground state
Excited state
ν
Light
Ground state
ΔE = hν
Why materials emit light?
You have to learn about interaction mechanism of matter with
electromagnetic field if you want to understand these phenomena
In the field of Quantum electronics
Light emission from materials
According to classical electromagnetics, Rutherford
atom model is unstable. It predicts lifetime of atoms
are order of 10-11 sec. (See the final subject in my
lecture note ElectromagneticsⅡ)
v
ω
Electron −e
m
Proton
r
+e
In order to solve this antinomy, Quantum mechanics
was proposed
N. Bohr proposed an atom model which electrons
exist as standing wave of matter-wave. The shape
of the standing wave is defined by the quantum
condition, and it is arrowed in several discreet
states. When an electron transits from one steady
state to other state, it emit / absorb photon which
energy correspond to energy difference between
the two states. (ΔE = hν)
Rutherford atom model
Proton
−e
+e
Which process occur ? Light emission or absorption ?
Why electrons make transition between the states ? Bohr hydrogen atom model
Theory describing light absorption and emission
There are three methods describing interaction of matter with electromagnetic fields
1. Classical theory
2. Semi-classical theory
Energy of electrons in semiconductor is quantized (Band structure),
On the other hand, energy of electromagnetic field is treated by
classical electromagnetics
3. Quantum theory
Energy of electrons in semiconductor is quantized (Band structure),
Electromagnetic field is also quantized (Field quantization)
Three methods and their applicable phenomena
Electromagnetic Optical
field
absorption
Stimulated Spontaneous
emission
emission
Method
Matter
Classical theory
Classical
Classical
Possible
Impossible
Impossible
Semi-classical theory Quantum
Classical
Possible
Possible
Impossible
Quantum
Possible
Possible
Possible
Quantum theory
Quantum
Description of electromagnetic field
In order to understand interaction of matter with electromagnetic field, we need to
describe electromagnetic field and to understand its fundamental characteristics
Maxwell equations
B( x, t )
t
E ( x, t )
1 E ( x, t )
  B( x, t )  
 ie  2
 ie
t
v
t
  E ( x, t )  
  E ( x, t ) 
e

  B( x, t )  0
Electric field E and magnetic field B can be also described as follows with
electromagnetic potential A(x, t) and ϕ(x, t)
E 
A
  ,
t
B   A
Therefore, Maxwell equations can be replaced to equations with A and ϕ as
values of electromagnetic field, instead of E and B
Description of electromagnetic field
Electromagnetic potentials can be described as follows with arbitrary scalar
function χ(x, t).
AL ( x, t )  A( x, t )  grad  ( x, t ),
L ( x, t )   ( x, t ) 
 ( x, t )
t
This function χ(x, t) is called a “gauge function”, and selecting these new
electromagnetic potential AL and ϕL is called “gauge transformation”.
When χ(x, t) was selected as AL and ϕL satisfying the following relation,
div AL ( x, t )  
L ( x, t )
0
t
the gauge is called “Lorenz gauge”. In this case, basic equations that describe
electromagnetic phenomena is reduced to two simple equations regarding AL
and ϕL as follows.

2 
    2  AL ( x, t )    ie ( x, t ),
t 


2 
1
    2 L ( x, t )   e ( x, t )
t 


These equations indicate that electromagnetic potential AL and ϕL caused by ie
or e propagate as wave with light speed.
Description of electromagnetic field
Other than Lorenz gauge, when A was selected as satisfying   A  0 condition,
the gauge is called “Coulomb gauge”.
In this case, basic equations that describe electromagnetic phenomena is as follows
  



2 

    2  A      i
t 
t

In free space where both electric charge ρe and electric current ie do not exist,

2 
  Const.,     2  A  0
t 

Therefore, E and B is derived from A with the following relations
A
t
B   A
E 
By selecting Coulomb gauge, electromagnetic fields can be
described by only vector potential A, because scalar
potential ϕ is constant in whole space when electric charge
dose not exist in the thinking space.
Interaction of charged particles with electromagnetic fields
When single charged particle (electron) is in electromagnetic field,
Hamiltonian of the particle is described as follows
H
1
( p  eA) 2  eV
2m
Here, p is the momentum operator, m is electron mass, V is potential of
electron, e is elementary charge, and A is vector potential. Hamiltonian H
can be also written as,
1
H
( p  eA) 2  eV  H 0  H int
2m
1 2
H0 
p  eV
2m
e
e2 2
H int  
( p  A  A  p) 
A
2m
2m
Here, H0 is an Unperturbed Hamiltonian which is for an electron in space
without electromagnetic field, and Hint is an Interaction Hamiltonian which is
originated by interaction between an electron and electromagnetic field.
The last term in Hint is proportional to A2, and it reveals higher order effects
(nonlinear optical effect). Here, we ignore it because the contribution is small.
Electrical-dipole approximation of the interaction
Position of a charged particle is described as
r  r0eit  r0*eit
Momentum of a charged particle is described as
Polarization
r of electron
cloud
E
p  mr  m(ir0eit  ir0*eit )
Vector potential is described as
Polarization of atom
A  A0eit  A0*eit
E
Electric field is
E
+e
e+iωt
A
 iA0 eit  iA0*e it
t
Therefore,
r
e−iωt
−e
Electrical dipole
e
e
( p  A  A  p)   m(i r0  A0* i r0*  A0 )
2m
m
 e r  E
 R  E
H int  
Here, R = er, and it called electric dipole moment
Terms “ei2ωt ” or
“e−i2ωt ” disappear
when integrating
for time
Electrical-dipole approximation of the interaction
Interaction Hamiltonian
e
H int  
( p  A  A  p)
2m
only include effect by electric field “RE” although the derived interaction
equation for a charged particle include effects by both electric field (Coulomb
force) and magnetic field (Lorentz force). The reason is that we assume the
motion of a charged particle as r  r0eit  r0*eit which is vibration at a limited
place. If we assume parallel motion for it, effect by magnetic field will be also
included.
In this way, when the interaction only depend on electric field, and the interaction
can be described as RE, it is called electric dipole approximation.
In some case, higher order polarization “multipolar” (such as electric quadrupole
or electric octopole) occasionally emerge.
皆さんへのメッセージ
工学で扱う物理現象は非常に複雑なものが多い。従って、全ての物理現象を取り入
れた完璧な理論を構築することは不可能である。
良きエンジニアとは、それら複雑な物理現象の中で、何が本質的に重要かを見極め、
近似をうまく使い、無視できる物理現象は思い切って無視し、シンプルな理論を構築
できる人である。ただし、どんな近似を使ったのかは決して忘れてはいけない。
Electrical dipole in semiconductor
In semiconductor, an electron and hole pair forms a electrical dipole
Conduction
band
−e
Electron
+
Electron
Electrical dipole
E
Electrical dipole
+e
Hole
Hole
Valence band
Electrical dipole in semiconductor
Quantum statistics and density matrix
Physical quantities involving many particles (electrons) such as electrical
current are statistical average values for the particles. Furthermore, expected
values for multiple measurements of a single event is needed statistical
treatments. We discuss about statistical nature for many particles or multiple
measurements. State for ν th particle or ν th measurement can be described as
 ( )   Cn( ) n .
n
Here, n is an energy eigenstate for single particle. It is assumed to form
complete space. Therefore, any  ( ) can be formed by linear combination of n .
n does not required the suffix (ν).
If operator for a physical quantity is assumed as A, expected value for ν th particle
is
 ( ) A  ( )   Cn( )*Cm( ) n A m .
n,m
Next, we develop an average of expected values for the group of particle
(ensemble average). We assume the contribution from the ν the particle
(probability for finding n th particle) as P(ν) , and normalize it.
P


( )
1
Quantum statistics and density matrix
Statistical average (ensemble average) of the expected value is described as
 A    P ( )  ( ) A  ( ) 

P C


( )
Cm( ) n A m
( )*
n
,n,m
Here, we rewrite as below
A   A  , Anm  n A m , mn   P( )Cn( )*Cm( )  Cn*Cm

 11

   21



Matrix ρ having ρmn as its elements is called density matrix
Using density matrix
A    mn Anm   m  n n A m
n ,m
n ,m
12  
 22  


 

 


  m   n n  A m   m A m .
m
m
n

 I  n n
 m A m
m
is identity operator
n
is summation of on-diagonal elements of ρA, that is Trace.
A   m A m  T r( A) , and T r(  )  1
m
Motion equation of density matrix
( )
密度行列 ρ の性質が分かれば、集団内の個々の粒子についての状態 
や抽出
確率 P(ν) を知らなくても、統計性を含めた期待値 A を知ることができる。そこで、密度
行列 ρ を表現する方程式、つまり ρ が従うべき方程式を求めてみる。
まず、密度行列の行列要素の定義式を、以下のように書き直す。
 mn   P ( )Cn( )*Cm( )   P ( )  ( ) n m  ( )




  m  ( ) P ( )  ( ) n  m   ( ) P ( )  ( )  n



( )
( )
( )
従って密度行列は、     P 
と書くことができる。

この式の両辺を時間 t で微分すると、
( )

d  ( )
d
d 
( )
( )
( )
( )
 
P   P
dt
dt
dt
 






ただし、抽出確率 P(ν) の時間依存性はないとしている。
となる。
Motion equation of density matrix
Schroedinger equation i
 i
d
 ( )   ( ) H
dt
d ( )

 H  ( ) and its Hermitian conjugate
dt
より、
d
H
H
    ( ) P ( )  ( )   ( ) P ( )  ( )

dt
i 
  i
1
1
 H  H   H ,  
となる。
i
i
これは、密度行列の時間発展を表す式であり、密度行列の運動方程式或いは、
量子リウヴィル(Liouville)方程式もしくはリウヴィル-フォン・ノイマン方程式とも呼
ばれる。
双極子との相互作用がある場合の密度行列
電子系の主ハミルトニアンを H0 とし、電気双極子能率を R とする。前に述べたように、
電場が存在するときの相互作用ハミルトニアン Hint は、−RE となり、電子系全体のハ
ミルトニアン H は、
H  H0  Hint  H0  RE
となる。
これを、密度行列の運動方程式に代入すると、
d 1
1
1
 H  H   H 0   H 0   RE  RE 
dt i
i
i
1
1
 H 0   H 0   R  R E
となる。
i
i
ここで、最後の ≈ では、１個の電子が存在する領域が電磁波の波長に比べて十分に
小さく、その範囲内で電場の分布は一定と見なせることを仮定している。実際、気体
原子に束縛されている電子の存在範囲はせいぜい数Å程度であり、また半導体中の
電子に至ってもせいぜい数十Å程度である。それに対して、相互作用する光の波長
は数千Åもあるので、この仮定は妥当である。
双極子との相互作用がある場合の密度行列
次に、密度行列の行列要素に対する方程式を導出する。エネルギー固有状態には
時間依存性が無いので、
d mn d
d

mn  m
n
dt
dt
dt
と表すことができ、従って、
d mn 1
1
1
1

m H0 n 
m H 0 n 
m R n E 
m R n E
dt
i
i
i
i
1
1





m H 0   l l  n 
m    l l H 0 n
i
i
 l

 l

1
1





m    l l R n E 
m R  l l   n E
i
i
 l

 l

1
1
  m H0 l l  n   m  l l H0 n
i l
i l
1
1
  m  l l R n E  m Rl l  n E
i l
i l
と書くことができる。なお。ここで用いている固有状態は、主ハミルトニアン H0 の固有
状態である。
双極子との相互作用がある場合の密度行列
従って、状態 n は、エネルギー固有値 Wn をとり、状態間には直交性があるので、
d mn 1
1
1
1
 Wm m  n 
m  n Wn   m  l l R n E   m R l l  n E
dt
i
i
i l
i l
1
1
1
  mn Wm  Wn     ml Rln E   Rml ln E
i
i l
i l
1
 i mn nm    ml Rln  Rml  ln E
となる。
i l
ただし ωmn は、順位 m と順位 n とのエネルギー差に対応する角周波数であり、
Wn  Wm  nm
で与えられる。

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