1. Communications Devices
Lecture on 6/21, 6/28, 7/5, 7/12, 7/19, 7/26 in 2016
Hirohito YAMADA

2. About lecture 1. Schedule
　　6/21 Backbroung of this lecture, Basic of semiconductor photonic devices 　　6/28 Matter-electromagnetic wave interaction based on semi-classical theory 　　7/5 Optical amplification and Laser 　　7/12 Electromagnetic field quantization and quantum theory
　　7/19 Optical transition in semiconductor, Photo diode, Laser diode 　　7/26 Optical amplifier, Optical modulator, Optical switch,
Optical wavelength filter, and Optical multiplexer/demultiplexer　
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　 or ECEI 2nd Bld. Room 203
4. Lecture note dounload　URL:

3. Growth of internet traffic in Japan
Total download traffic in Japan was about 2.6T bps at the end of 2014
year
2005
2006
2007
2008
2009
2010
2011
2004
2013
2012
Daily average value
Annual growth rate: 30%
Total download traffic in Japan
Total upload traffic in Japan
Cited from: H27年度版情報通信白書

4. Optical fiber submarine networks
Cited from

5. Power consumption forecast of network equipments
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
Annual power consumption of network equipments (×1011 Wh)
year
Total internet traffic (Tbps)
Network traffic
Total power generation at 2007
Power consumption

6. 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

7. 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
- 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)
: Treated in our lecture
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.)

8. 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”

9. What is Photonic Device ?
Manipulating electron charge
Electronics
voltage and current
Manipulating wavefunction of electron
Electron Tube, Diode, Tr, FET, LSI
Tunnel effect devices
Magnetic Recording
Photo-electronics
(Opt-electronics)
Manipulating both spin and charge of electrons
Display
Spintronics
CCD, CMOS sensor
Manipulating photon and electron charge
GMR HDD
MRAM ?
solar cell
LD, LED
Photo detector, PD
Unexplored
Manipulating photon
Magnetics
(Spinics)
Photonics
Opto-spinics
Energy and number
Laser
magnet-optical disk
HDD
Optical disk
Electromagnetic wave
Magnetic tape
Optical fiber
Manipulating spin of materials
amplitude and phase

10. Basis of semiconductor photonic devices
Properties required for semiconductor used as photonic devices
Passive devices(Non light-emitting devices)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Wavelength: λ (μm)
Optical absorption coefficient: α (cm-1)
T = 300K
105
104
103
102 Si
InP
GaAs Ge
In0.53Ga0.47As
・ 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 of major semiconductor materials

11. 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
In bulk Si, holes distribute at around the Γ point, on the other hand, electrons distribute at around the X point (Indirect transition semiconductors)
Hole
Band gap ~1.1eV
Dispersion relation of electron energy in Si

12. 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

13. 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
1.6%
tensile strain
Conduction band
Valence band
Indirect transition
Direct transition
Band structure of Ge

14. 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
Real part
Imaginary part
Photon energy (eV)
Calc.
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.
W. Sellmeier equation
Phenomenologically-derived equation of relation between wavelength and refractive index
Here, λi = c/νi, c: light speed, νi: resonance frequency of material, Ai: Constant
Calculation of dielectric function ε of Si

15. Basis of semiconductor photonic devices
Birefringence
In anisotropic medium, dielectric constant (refractive index), magnetic permeability are tensor
Optical axis
Outgoing ray
(Material equation)
extraordinary ray
Incident ray
Crystal is optically anisotropic medium
ordinary ray
When light beam enter a crystal, it splits two beams (ordinary ray and extraordinary 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

16. 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
Material equation
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.
Linear polarization
c : Electric susceptibility
When intensity of incident light become strong, electric susceptibility become depended on the electric field E

17. Basis of semiconductor photonic devices
Electron transition in direct transition type and indirect transition type semiconductors
Band structure of (a) GaAs and (b) Si
In case of indirect transition, phonon intervenes light emission or absorption

18. 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
Ground state
Light from burning materials ‥‥ Chemical reaction of organic materials
Excited state ν
Electroluminescence from LED‥‥Electron transition in semiconductors ΔE
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

19. Light emission from materials
According to classical electromagnetics, Rutherford atom model is unstable. It predicts lifetime of atoms are order of sec. (See the final subject in my lecture note ElectromagneticsⅡ) +e −e
Electron
Proton
Rutherford atom model m r v ω
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) −e
Proton +e
Which process occur ? Light emission or absorption ?
Why electrons make transition between the states ?
Bohr hydrogen atom model

20. 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
Classical theory
Semi-classical theory
Quantum theory
Electromagnetic field
Matter
Method
Optical absorption
Possible
Stimulated emission
Impossible
Spontaneous emission
Classical
Quantum

21. 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
Electric field E and magnetic field B can be also described as follows with electromagnetic potential A(x, t) and ϕ(x, t)
Therefore, Maxwell equations can be replaced to equations with A and ϕ as values of electromagnetic field, instead of E and B

22. Description of electromagnetic field
Electromagnetic potentials can be described as follows with arbitrary scalar function χ(x, 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,
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.
These equations indicate that electromagnetic potential AL and ϕL caused by ie or e propagate as wave with light speed.

23. Description of electromagnetic field
Other than Lorenz gauge, when A was selected as satisfying condition,
the gauge is called “Coulomb gauge”.
In this case, basic equations that describe electromagnetic phenomena is as follows
In free space where both electric charge ρe and electric current ie do not exist,
Therefore, E and B is derived from A with the following relations
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.

24. Interaction of charged particles with electromagnetic fields
When single charged particle (electron) is in electromagnetic field, Hamiltonian of the particle is described as follows
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,
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.

25. Electrical-dipole approximation of the interaction
Position of a charged particle is described as
Polarization of electron cloud
Momentum of a charged particle is described as E r
Vector potential is described as
Polarization of atom
e+iωt +e E
Electric field is
e−iωt r −e
Therefore,
Electrical dipole
Terms “ei2ωt ” or “e−i2ωt ” disappear when integrating for time
Here, R = er, and it called electric dipole moment

26. Electrical-dipole approximation of the interaction
Interaction Hamiltonian
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 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。

27. Electrical dipole in semiconductor
In semiconductor, an electron and hole pair forms a electrical dipole
Conduction band
Electron
Electron −e
Electrical dipole E
Electrical dipole + +e
Hole
Hole
Valence band
Electrical dipole in semiconductor

28. 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 .
Here, 　　is an energy eigenstate for single particle. It is assumed to form complete space. Therefore, any can be formed by linear combination of
　 does not required the suffix (ν).
If operator for a physical quantity is assumed as A, expected value for ν th particle is .
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.

29. Quantum statistics and density matrix
Statistical average (ensemble average) of the expected value is described as
Here, we rewrite as below
Matrix ρ having ρmn as its elements is called density matrix
Using density matrix .
is identity operator
is summation of on-diagonal elements of ρA, that is Trace.
, and

30. Motion equation of density matrix
If we know the nature of the density matrix ρ, we can obtain expected value including statistical nature, even we do not know the state of each particle in the group or the probability for finding n th particle P(ν). Then we direct equations for density matrix.
Here, we rewrite the definition of matrix elements of density matrix as below
Then, density matrix can be described as
By differentiating both sides of this equation with respect to t, we can obtain
Here, we assume probability for finding n th particle P(ν) is time independent.

31. Motion equation of density matrix
From Schroedinger equation 　　　　　　and its Hermitian conjugate
, we obtain .
This denotes time evolution of density matrix. Then we call it motion equation of density matrix or Quantum Liouville equation.