ELEC 425/ 6261 Lecture Notes 2 & 3

ELEC 425/ 6261Lecture Notes 2 & 3

Optical waveguides display 3 types of dispersion:

These are the main sources of dispersion in the fibers.

• Material dispersion, different wavelength of light travel at different velocities within a given medium.Due to the variation of n1 of the core wrtwavelength of the light.

•Waveguide dispersion, β depends on the wavelength, so even within a single mode different wavelengths will propagate at slightly different speeds.Due to the variation of group velocity wrt V-number

τt

Spread, ²τ

t0

λ

Spectrum, ²λ

λ1λ2λo

Intensity Intensity Intensity

Cladding

CoreEmitterVery shortlight pulse

vg(λ2)vg(λ1)

Input

Output

All excitation sources are inherently non-monochromatic and emit within aspectrum, ²λ, of wavelengths. Waves in the guide with different free spacewavelengths travel at different group velocities due to the wavelength depeof n1. The waves arrive at the end of the fiber at different times and hence a broadened output pulse.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

λτΔ=

Δ DL

Optical waveguides display 3 types of dispersion

• Modal dispersion, in waveguides with more than one propagating mode. Modes travel with different group velocities.Due to the number of modes traveling along the fiber with different group velocity and different path.

Low order modeHigh order mode

Cladding

Core

Light pulse

t0 t

Spread, Δτ

Broadenedlight pulse

IntensityIntensity

Axial

Schematic illustration of light propagation in a slab dielectric waveguideentering the waveguide breaks up into various modes which then propaggroup velocities down the guide. At the end of the guide, the modes comconstitute the output light pulse which is broader than the input light pul


0

1.2 1.3 1.4 1.5 1.61.1-30

20

30

10

-20

-10

λ (μm)

Dm

Dm + Dw

Dwλ0

Dispersion coefficient (ps km -1 nm-1)

Material dispersion coefficient (Dm) for the core material (taken asSiO2), waveguide dispersion coefficient (Dw) (a = 4.2 μm) and thetotal or chromatic dispersion coefficient Dch (= Dm + Dw) as afunction of free space wavelength, λ.


λτΔ=

ΔmD

L

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈ 2

2

λλ

dnd

cDm

λτω Δ=

Δ DL

( ) 22

22

22

984.1

cna

ND g

πω ≈

λτΔ=

ΔPD

Lλτ

ω Δ++=Δ

pm DDDL

Material Dispersion Coefficient

Waveguide Dispersion Coefficient

Core

z

n1x // x

n1y // y

Ey

Ex

Ex

Ey

E

Δτ = Pulse spread

Input light pulse

Output light pulset

t

Δτ

Intensity

Suppose that the core refractive index has different values along two orthogonaldirections corresponding to electric field oscillation direction (polarizations). We cantake x and y axes along these directions. An input light will travel along the fiber with Exand Ey polarizations having different group velocities and hence arrive at the output atdifferent times


Polarization Dispersion

Material and waveguide dispersion coefficients in anoptical fiber with a core SiO2-13.5%GeO2 for a = 2.5to 4 μm.

0

–10

10

20

1.2 1.3 1.4 1.5 1.6–20

λ (μm)

Dm

Dw

SiO2-13.5%GeO2

2.53.03.54.0a (μm)

Dispersion coefficient (ps km-1 nm-1)


20

-10

-20

-30

10

1.1 1.2 1.3 1.4 1.5 1.6 1.7

0

30

λ (μm)

Dm

Dw

Dch = Dm + Dw

λ1

Dispersion coefficient (ps km -1 nm-1)

λ2

n

r

Thin layer of claddingwith a depressed index

Dispersion flattened fiber example. The material dispersion coefficient (Dm) for thecore material and waveguide dispersion coefficient (Dw) for the doubly clad fiberresult in a flattened small chromatic dispersion between λ1 and λ2.


t0

Emitter

Very shortlight pulses

Input Output

Fiber

PhotodetectorDigital signal

Information Information

t0

~2² τ1/2

T

t

Output IntensityInput Intensity² τ1/2

An optical fiber link for transmitting digital information and the effect ofdispersion in the fiber on the output pulses.


n1

n2

21

3

nO

n1

21

3

n

n2

OO' O''

n2

(a) Multimode stepindex fiber. Ray pathsare different so thatrays arrive at differenttimes.

(b) Graded index fiber.Ray paths are differentbut so are the velocitiesalong the paths so thatall the rays arrive at thesame time.

23


0.5P

O'O

(a)

0.25P

O

(b)

0.23P

O

(c)

Graded index (GRIN) rod lenses of different pitches. (a) Point O is on the rod facecenter and the lens focuses the rays onto O' on to the center of the opposite face. (b)The rays from O on the rod face center are collimated out. (c) O is slightly away fromthe rod face and the rays are collimated out.


z

A solid with ions

Light directionk

Ex

Lattice absorption through a crystal. The field in the waveoscillates the ions which consequently generate "mechanical"waves in the crystal; energy is thereby transferred from the waveto lattice vibrations.


Sources of Loss and Attenuation in Fibers

Scattered waves

Incident wave Through wave

A dielectric particle smaller than wavelength

Rayleigh scattering involves the polarization of a small dielectricparticle or a region that is much smaller than the light wavelength.The field forces dipole oscillations in the particle (by polarizing it)which leads to the emission of EM waves in "many" directions sothat a portion of the light energy is directed away from the incidentbeam.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Escaping wave

θ θ

θ′ < θ

θθ > θc θ′

Microbending

R

Cladding

Core

Field distribution

Sharp bends change the local waveguide geometry that can lead to wavesescaping. The zigzagging ray suddenly finds itself with an incidenceangle θ′ that gives rise to either a transmitted wave, or to a greatercladding penetration; the field reaches the outside medium and some lightenergy is lost.


Attenuation in Optical Fiber

)10(

)log(101

10/Linout

out

indB

PP

PP

Lα

αα

−=

==G. Keiser (Ref. 1)

0.05

0.1

0.51.0

5

10

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Latticeabsorption

Rayleighscattering

Wavelength (µm)

Illustration of a typical attenuation vs. wavelength characteristicsof a silica based optical fiber. There are two communicationschannels at 1310 nm and 1550 nm.

OH-absorption peaks

1310 nm

1550 nm


Some Concepts of Solid State MaterialsContents

The Semiconductors in EquilibriumNonequilibrium Condition

Generation-RecombinationGeneration-Recombination rates

Photoluminescence & ElectroluminescencePhoton Absorption

Photon Emission in SemiconductorsBasic Transitions

RadiativeNonradiative

Spontaneous EmissionStimulated Emission

Luminescence EfficiencyInternal Quantum EfficiencyExternal Quantum Efficiency

Photon AbsorptionFresnel Loss

Critical Angle LossEnergy Band Structures of Semiconductors

PN junctionsHomojunctions, Heterojunctions

MaterialsIII-V semiconductors

Ternary SemiconductorsQuaternary SemiconductorsII-VI SemiconductorsIV-VI Semiconductors

The Semiconductors in Equilibrium

• The thermal equilibrium concentration of carriers is independent of time.

• The random generation-recombination of electrons-holes occur continuously due to the thermal excitation.

• In direct band-to-band generation-recombination, the electrons and holes are created-annihilated in pairs:

• Gn0 = Gp0 , Rn0 = Rp0

• -The carriers concentrations are independent of time therefore:

• Gn0 = Gp0 = Rn0 = Rp0

Nonequilibrium Conditions in Semiconductors

• When current exist in a semiconductor device, the semiconductor is operating under nonequilibrium conditions. In these conditions excess electrons in conduction band and excess holes in the valance band exist, due to the external excitation (thermal, electrical, optical…), in addition to thermal equilibrium concentrations.

n(t) = no + δn(t), p(t) = po + δp(t)

• The behavior of the excess carriers in semiconductors (diffusion, drift, recombination, …) which is the fundamental to the operation of semiconductors (electronic, optoelectronic, ..) is described by the ambipolar transport equations.

Generation-Recombination

• In the direct band-to-band the excess electrons and holes are generated and recombined in pair:

g’n0 = g’p0 , R’n0 = R’p0

Generation-Recombination Rates

• The recombination rate is proportional to electron and hole concentrations.

is the thermal equilibrium generation rate.

dttnd

dttnndtptnn

dttdn

i)())(()]()([)( 02 δδα =

+=−=

)()( 0 tnntn δ+= )()( 0 tpptp δ+=

2inα


• Electron and holes are created and recombined in pairs, therefore,

• δn(t) = δp(t) and no and po are independent of time.

)]())[(()]())((([))((0000

2 tnpntntpptnnndt

tndi δαδδδαδ

++−=++−=

Considering a p-type material under low-injection condition,

)())((0 tnp

dttnd δαδ

−= 00 )0()0()( n

ttp enentn τα δδδ

−− ==

τno = (αpo)-1 is the minority carrier electrons lifetime, constant for low-injections.


• The recombination rate ( a positive quantity) of excess minority carriers (electrons-holes) for p-type materials is:

0

'' )(

npn

tnRRτ

δ==

Similarly, the recombination rate of excess minority carriers for n-type material is:

0

'' )(p

pntnRR

τδ

==


• where τpo is the minority carrier holes lifetime.

• For high injections which is in the case of LASER and LED operations, δn >> no and δp >> po so,

τno = (αδp)-1 and τpo = (αδn)-1

• If photons are emitted (usually in direct bandgapsemiconductors), the process is called radiative(important for the operation of optical devices), otherwise is called nonradiative recombination (takes place via surface or bulk defects and traps).


• In any carrier decay process the total lifetime τ can be expressed as

nrr τττ111

+=

where τr and τnr are the radiative and nonradiative lifetimes respectively.

The total recombination rate is given by

Where Rr and Rnr are radiative and nonradiative recombination rates per unit volume respectively and Rsp is called the spontaneous recombination rate.

spnrrtotal RRRR =+=

Photoluminescence & Electroluminescence

• The recombination of excess carries in direct bandgapsemiconductors may result in the emission of photon. This property is generally referred to as luminescence.

• If the excess electrons and holes are created by photon absorption, then the photon emission from the recombination process is called photoluminescence.

• If the excess carries are generated by an electric current, then the photon emission from the recombination process is called electroluminescence.

Absorption

• When semiconductors are illuminated with light, the photons may be absorbed (for Eph = hν≥Eg= E2 – E1)or they may propagate through the semiconductors (for Eph≤Eg).

• There is a finite probability that electrons in the lower level absorb energy from incoming electromagnetic field (light) with frequency of ν ≥ (E2 – E1)/h and jump to the upper level.

• B12 is proportionality constant, δν = ν2 - ν1 and Φδν = Iνis the photon density in the frequency range of δν.

1121 )( NB

dtdN

ab

δνΦ−=

Absorption

• dN1 states are raised from E1 to E2 i.e dN1 photons are absorbed.• Electrons are created in conduction band and holes in valence

band. • When photons with an intensity of Iν (x) are traveling through a

semiconductor, going from x position to x + dx position (in 1-D system), the energy absorbed by semiconductor per unit of time is given by αIν (x)dx, where α is the absorption coefficient; the relative number of photons absorbed per unit distance (cm-1).

dxxIdxdx

xdIxIdxxI )(.)()()( νν

νν α−==−+

)()(

xIdx

xdIν

ν α−=

xeIxI ανν

−= 0)( where Iν(0) = Iν0 .

Absorption

• Intensity of the photon flux decreases exponentially with distance.• The absorption coefficient in semiconductor is strong function of

photon energy and band gap energy.• The absorption coefficient for hν < Eg is very small, so the

semiconductor appears transparent to photons in this energy range.

Photon Emission in Semiconductors

• When electrons in semiconductors fall from the conduction band to the valence band, called recombination process, release their energy in form of light (photon), and/or heat (lattice vibration, phonon).

• N1 and N2 are the concentrations of occupied states in level 1 (E1) and level 2 (E2) respectively.

Basic Transitions

RadiativeIntrinsic emissionEnergetic carriers

NonradiativeImpurities and defect center involvement

Auger process







Einstein Relationship

Einstein Relationship

Wave attenuation

pn Junction

The entire semiconductor is a single-crystal material:

-- p region doped with acceptor impurity atoms

--n region doped with donor atoms

--the n and p region are separated by the metallurgical junction.

-eNa

eNd

x

B-

h+

p n

M

As+

e-

W

Neutral n-regionNeutral p-region

Space charge region

Metallurgical Junction(a)

(b)

-xp

xn(c)

E E (x)

eVbi

e φ (x)

x

x0

Eo

M

xn

xn

- xp

- xp

(e)

(f)

(d)

ρ (x)

Hole PE(x)

Electron PE(x)

- the potential barrier :

- keeps the large concentration of electrons from flowing from the n region into the p region;

- keeps the large concentration of holes from flowing from the p region into the n region;

=> The potential barrier maintains thermal equilibrium.

- the potential of the n region is positive with respect to the p region => the Fermi energy in the n region – lower than the Fermi energy in the p region;

- the total potential barrier – larger than in the zero-bias case;

- still essentially no charge flow and hence essentially no current;

-a positive voltage is applied to the p region with respect to the n region;

- the Fermi energy level – lower in the p region than in the n region;

- the total potential barrier – reduced => the electric field in the depletion region – reduced;

⇒ diffusion of holes from the p region across the depletion region into the n region;

⇒ diffusion of electrons from the n region across the depletion region into the p region;

- diffusion of carriers => diffusion currents;

⎟⎟⎠

⎞⎜⎜⎝

⎛= 2ln

i

datbi n

NNVV

-in thermal equilibrium :

- the n region contains many more electrons in the conduction band than the p region;

- the built-in potential barrier prevents the large density of electrons from flowing into the p region;

- the built-in potential barrier maintains equilibrium between the carrier distribution on either side of the junction;

⎟⎠⎞

⎜⎝⎛ −

=kTqVnn bi

np exp00

- the electric field Eapp induced by Va – in opposite direction to the electric field in depletion region for the thermal equilibrium;

- the net electric field in the depletion region is reduced below the equilibrium value;

- majority carrier electrons from the n side -> injected across the depletion region into the p region;

- majority carrier holes from the p region -> injected across the depletion region into the n region;

- Va applied => injection of carriers across the depletion regions-> a current is created in the pn junction;

2/1

max

2/1

max

2/1

2

)(2)(2

)(2

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+

−≅

>>

+−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+

+−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡ ++=+=

da

da

s

R

biR

Rbi

da

da

s

Rbi

da

daRbispn

NNNNeVE

VV

WVV

NNNNVVeE

NNNN

eVVxxW

ε

ε

ε

212

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=+=

da

dabisnp NN

NNeVxxW ε For zero bias

For reverse biased

WVE bi2

max−

=

For reverse biased

For zero bias

When there is no voltage applied across the pn junction the junction is in thermal equilibrium => the Fermi energy level –constant throughout the entire system.

FpFnbiV Φ+Φ=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

p

n

n

p

i

dabi n

ne

kTpp

ekT

nNN

ekTV lnln 2

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=−+−=Φ+Φ=

i

d

i

aFiFnFpFinpbi n

Ne

kTnN

ekTeEEeEEV lnln/)(/)(

dxxdEx

dxxd

EquationsPoisson

s

)()()(

'

2

2−=

−=

ερφ

+

_

ρ(C/cm³)deN+

aeN−

nx+px−

E

x =0

∫−=x

xs pdxxxE )(1)( ρ

ε

nns

d

pps

a

xxxxeNE

xxxxeNE

≤≤−−

=

≤≤−+−

=

0),(

0),(

ε

ε

cmFSiFor rs /)1085.8)(7.11( 140

−×=⋅= εεε

ndpa xNxN =

Charge neutrality:

The peak electric field Is at x = 0

s

pa

s

nd xeNxeNEεε

−=−=max

nx+px−

Emax

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

n

pn

p

nps L

nqDL

pqDJ 00

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛= 1exp

kTqVJJ a

s - ideal-diode equation;

The bipolar transistor:

- tree separately doped regions

- two pn junctions;

The width of the base region – small compared to the minority carrier diffusion length;

The emitter – largest doping concentration;

The collector – smallest doping concentration;

- the bipolar semiconductor – not a symmetrical device;

-the transistor – may contain two n regions or two p regions -> the impurity doping concentrations in the emitter and collector = different;

-- the geometry of the two regions –can be vastly different;

Electromagnetic Spectrum

• Three basic bands; infrared (wavelengths above 0.7μm), visible (wavelengths between 0.4-0.7μm), and ultraviolet light (wavelengths below 0.4μm).

• E = hν = hc/ λ ; c = νλ

• An emitted light from a semiconductor optical device has a wavelength proportional to the semiconductor band-gap.

• Longer wavelengths for communication systems; Eg ≈ 1μm. (lower Fiber loss).

• Shorter wavelengths for printers, image processing,… Eg > 1μm.

• Semiconductor materials used to fabricate optical devices dependon the wavelengths required for the operating systems.

Materials

• Almost all optoelectronic light source depend upon epitaxial crystal growth techniques where a thin film (a few microns) of semiconductor alloys are grown on single-crystal substrate; the film should have roughly the same crystalline quality. It is necessary to make strain-free heterojunction with good-quality substrate. The requirement of minimizing strain effectsarises from a desire to avoid interface states and to encourage long-term device reliability, and this imposes a lattice-matching condition on the materials used.

Materials

• The constraints of bandgap and lattice match force that more complex compound must be chosen. These compounds include ternary (compounds that containing three elements) and quaternary (consisting of four elements) semiconductors of the form AxB1-xCyD1-y; variation of x and y are required by the need to adjust the band-gap energy (or desired wavelength) and for better lattice matching. Quaternary crystals have more flexibility in that the band gap can be widely varied while simultaneously keeping the lattice completely matched to a binary crystal substrate. The important substrates that are available for the laser diode technology are GaAs, InP and GaP. A few semiconductors and their alloys can match with these substrates. GaAs was the first material to emit laser radiation,and its related to III-V compound alloys, are the most extensively studied developed.

Materials

• III-V semiconductors• Ternary Semiconductors; Mixture of binary-binary

semiconductors; AxB1-xC; mole fraction, x, changes from 0 to 1.(x will be adjusted for specific required wavelength).

• GaxAl1-xAs ; In0.53Ga0.47As; In0.52Al0.48As

- Vegard’s Law: The lattice constant of AxB1-xC varies linearly from the lattice constant of the semiconductor AC to that of the semiconductor BC.

- The bandgap energy changes as a quadratic function of x.

- The index of refraction changes as x changes.• The above parameters cannot vary independently

(due to the only one degree of freedom).

• Quaternary Semiconductors; AxB1-xCyD1-y • (x and y will be adjusted for specific wavelength and

matching lattices).

• GaxIn1-xPyAs1-y ; (AlxGa1-x)yIn1-yP; AlxGa1-xAsySb1-y

2cxbxaEg ++=

Materials

• II-VI Semiconductors• CdZnSe/ZnSe; visible blue lasers.• Hard to dope p-type impurities at

concentration larger than 2×1018cm-3 (due to self-compensation effect). Densities on this order are required for laser operation.

Materials

• IV-VI semiconductors• PbSe; PbS; PbTe• By changing the proportion of Pb atoms in

these materials semiconductor changes from n- to p-type.

• Operate around 50 Ko

• PbTe/Pb1-xEuxSeyTe1-y operates at 174 Ko

Materials

Materials

Materials

Materials

• In the near infrared region, the most important and certainly the most extensively characterized semiconductors are GaAs, AlAs and their ternary derivatives AlxGa1-xAs.

• At longer wavelengths, the materials of importance are InP and ternary and quaternary semiconductors lattice matched to InP. The smaller band-gap materials are useful for application in the long wavelength range. The III-V compounds of interest here are antimony (Sb)-bearing compounds and InAs and their ternary and quaternary.

Energy Band Structure of Semiconductors

Basic Semiconductor Luminescent Diode StructuresLEDs (Light Emitting Diode)

• Under forward biased when excess minority carriers diffuse into the neutral semiconductor regions where they recombine with majority carriers. If this recombination process is direct band-to-band process, photons are emitted. The output photon intensity will be proportional to the ideal diode diffusion current.

• In GaAs, electroluminescence originated primarily on the p-side of the junction because the efficiency for electron injection is higher than that for hole injection.

• The recombination is spontaneous and the spectral outputs have a relatively wide wavelength bandwidth of between 30 – 40 nm.

• λ = hc/Eg = 1.24/ Eg

Basic Semiconductor Luminescent Diode Structures

Basic Semiconductor Luminescent Diode Structures

hυ - Eg

Eg (b)

V

(a)

p n+

Eg

eVo

EF

p n+

Electron in CBHole in VB

Ec

Ev

Ec

Ev

EF

eVo

Electron energy

Distance into device

(a) The energy band diagram of a p-n+ (heavily n-type doped) junction without any bias.Built-in potential Vo prevents electrons from diffusing from n+ to p side. (b) The appliedbias reduces Vo and thereby allows electrons to diffuse, be injected, into the p-side.Recombination around the junction and within the diffusion length of the electrons in thep-side leads to photon emission.


Light output

Insulator (oxide)p

n+ Epitaxial layer

A schematic illustration of typical planar surface emitting LED devices. (a) p-layergrown epitaxially on an n+ substrate. (b) First n+ is epitaxially grown and then p regionis formed by dopant diffusion into the epitaxial layer.

Light output

pEpitaxial layers

(a ) (b )

n+Substrate Substrate

n+

n+

Metal electrode


Light output

p

Electrodes

LightPlastic dome

Electrodes

Domedsemiconductor

pn Junction

(a) (b) (c)

n+n+

(a) Some light suffers total internal reflection and cannot escape. (b) Internal reflectionscan be reduced and hence more light can be collected by shaping the semiconductor into adome so that the angles of incidence at the semiconductor-air surface are smaller than thecritical angle. (b) An economic method of allowing more light to escape from the LED isto encapsulate it in a transparent plastic dome.

Substrate


Ec

Ev

EN

(b) N doped GaP

Eg

(a) GaAs1-yPy

y < 0.45(a) Photon emission in a direct bandgap semiconductor. (b). GaP is anindirect bandgap semiconductor. When doped with nitrogen there is anelectron trap at EN. Direct recombination between a trapped electron at ENand a hole emits a photon. (c) In Al doped SiC, EHP recombination isthrough an acceptor level like Ea.

Ec

Ev

Ea

(c) Al doped SiC


0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6λ

1.7Infrared

GaAs1-yPy

In1-xGaxAs1-yPyAlxGa1-xAs

x = 0.43

Indirectbandgap

Free space wavelength coverage by different LED materials from the visible spectrum to theinfrared including wavelengths used in optical communications. Hatched region and dashedlines are indirect Eg materials.

In0.49AlxGa0.51-xP


2 eV

2 eVeVo

Holes in VB

Electrons in CB1.4 eV

No bias

Withforwardbias

Ec

EvEc

Ev

EFEF

(a)

(b)

(c)

(d)

pn+ p

ΔEc

GaAs AlGaAsAlGaAs

ppn+

~ 0.2 μm

AlGaAsAlGaAs

(a) A doubleheterostructure diode hastwo junctions which arebetween two differentbandgap semiconductors(GaAs and AlGaAs)

(b) A simplified energyband diagram withexaggerated features. EFmust be uniform.

(c) Forward biasedsimplified energy banddiagram.

(d) Forward biased LED.Schematic illustration ofphotons escapingreabsorption in theAlGaAs layer and beingemitted from the device.


GaAs

E

Ec

Ev

Carrier concentrationper unit energy

Electrons in CB

Holes in VBhυ

1

0

Eg

hυ1

hυ2

hυ3

CB

VBλ

Relative intensity

1

0λ

1λ

2λ

3

ΔλΔhυ

Relative intensity

(a) (b) (c) (d)

Eg + kBT

(2.5-3)kBT

1/2kBT

Eg1 2 3

2kBT

(a) Energy band diagram with possible recombination paths. (b) Energy distribution ofelectrons in the CB and holes in the VB. The highest electron concentration is (1/2)kBT aboveEc . (c) The relative light intensity as a function of photon energy based on (b). (d) Relativeintensity as a function of wavelength in the output spectrum based on (b) and (c).


V

2

1

(c)

0 20 40I (mA)0

(a)

600 650 7000

0.5

1.0

λ

Relativeintensity

24 nm

Δλ

655nm

(b)

0 20 40I (mA)0

Relative light intensity

(a) A typical output spectrum (relative intensity vs wavelength) from a red GaAsP LED.(b) Typical output light power vs. forward current. (c) Typical I-V characteristics of ared LED. The turn-on voltage is around 1.5V.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

(a) Surface emitting LED (b) Edge emitting LED

Doubleheterostructure

Light

Light


ElectrodeSiO2 (insulator)

Electrode

Fiber (multimode)

Epoxy resin

Etched wellDouble heterostructure

Light is coupled from a surface emitting LEDinto a multimode fiber using an index matchingepoxy. The fiber is bonded to the LEDstructure.

(a)

Fiber

A microlens focuses diverging light from a surfaceemitting LED into a multimode optical fiber.

Microlens (Ti2O3:SiO2 glass)

(b)


Schematic illustration of the the structure of a double heterojunction stripecontact edge emitting LED

InsulationStripe electrode

SubstrateElectrode

Active region (emission region)

p+-InP (Eg = 1.35 eV, Cladding layer)

n+-InP (Eg = 1.35 eV, Cladding/Substrate)

n-InGaAs (Eg = 0.83 eV, Active layer)

Currentpaths

L

60-70 μm

Light beam

p+-InGaAsP (Eg = 1 eV, Confining layer)

n+-InGaAsP (Eg = 1 eV, Confining layer) 12 3200-300 μm


Multimode fiberLens

(a)

ELED

Active layer

Light from an edge emitting LED is coupled into a fiber typically by using a lens or aGRIN rod lens.

GRIN-rod lens

(b)

Single mode fiberELED


L λλmλm - 1

Fabry-Perot etalon

Partially reflecting plates

Output lightInput light

Transmitted light

Transmitted light through a Fabry-Perot optical cavity.


( )20

max 1 RI

I−

=( )

( ) ( )kLRRRII incidentdtransmitte 22

2

sin411+−−

=

A

BL

M1 M2 m = 1

m = 2

m = 8

Relative intensity

υ

δυm

υm υm + 1υm - 1

(a) (b) (c)

R ~ 0.4R ~ 0.81 υf

Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflectedwaves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowedin the cavity. (c) Intensity vs. frequency for various modes. R is mirror reflectance andlower R means higher loss from the cavity.


Lm =⎟⎠⎞

⎜⎝⎛

2λ m=1,2,3….

( ) ( )KLRRI

Icavity 220

sin41 +−=

LASERS

• The sensitivity of most photosensitive material is greatly increased at wave-length < 0.7 μm; thus, a laser with a short wave-length is desired for such applications as printers and image processing. The sensitivity of the human eye range between the wavelengths of 0.4 and 0.8μm and the highest sensitivity occur at 0.555μm or green so it is important to develop laser in this spectral regime for visual applications.

• Laser-quality crystals are obtained only with lattice mismatches <0.01% relative to the substrate. It is generally true that semiconductor materials with larger band gaps show a tendency to have small lattice constants.

LASERS

• The semiconductor laser diode is a forward bias p-n junction. The structure appears to be similar to the LED as far as the electron and holes are concerned, but it is quite different from the point of view of the photons. Electrons and holes are injected into an active region by forward biasing the laser diode. At low injection, these electrons and holes recombine (radiative) via the spontaneous process to emit photons. However, the laser structure is so designed that at higher injections the emission process occurs by stimulated emission. As we will discuss, the stimulated emission process provides spectral purity to the photon output, provides coherent photons, and offers high-speed performance.

• The exact output spectrum from the laser diode depends both on the nature of the optical cavity and the optical gain versus wavelength characteristics.

•• Lasing radiation is only obtained when optical gain in the medium can

overcome the photon loss from the cavity, which requires the diode current I to exceed a threshold value Ith and gop>gth

LASER Diode Modes of Threshold Conditions

Lasing Conditions:Population Inversion

Fabry-Perot cavity

gain (of one or several modes) > optical loss

( ) zhvhvg

eIzI ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−Γ

−

=α)(

)0()(

I – optical field intensityg – gain coefficient in F.P. cavity

- effective absorption coefficientΓ – optical field confinement factor. (the fraction of optical power in the active layer)

−

α

In one round trip i.e. z = 2L gain should be > loss for lasing;During this round trip only R1 & R2 fractions of optical radiation are reflected from the two laser ends 1 & 2.

2

21

21⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=nnnn

R

LASER Diode Modes of Threshold Conditions

From the laser conditions: )()2( oILI = 12 =− Lje β ( )mL πβ 22 =

)0()0()2(_

2

21 IeRRILIgL

==⎥⎦

⎤⎢⎣

⎡−Γ α

21

2 1_

RRe

gL

=⎥⎦

⎤⎢⎣

⎡−Γ α

21

_ 1ln2RR

gL =⎟⎠⎞

⎜⎝⎛ −Γ α

endth RRLg ααα +=+=Γ

_

21

_ 1ln21

Lasing threshold is the point at which the optical gain is equal to the total loss αt

endtthg ααα +==Γ_

thth Jgg β=≥

β – is constant and depends on the specific device construction.

Thus the gain

Laser Diode Rate Equations

Rate Equations govern the interaction of photons and electrons in the active region.

phspRBn

dtd

τφφφ

−+= = stimulated emission + spontaneous emission - photon loss

φτ

BnnqdJ

dtdn

sp

−−= = injection - spontaneous recombination - stimulated emission

(shows variation of electron concentration n).

Variation of photon concentration:

The relationship between optical output and the diode drive current:

d - is the depth of carrier-confinement regionB - is a coefficient (Einstein’s) describing the strength of the optical absorption and emission interactions,;Rsp - is rate of spontaneous emission into the lasing mode; τph – is the photon lifetime;τsp – is the spontaneous recombination lifetime; J – is the injection current density;


Solving the above Equations for a steady-state condition yields an expression for the output power.

0=dtd φ 0=

dtdnSteady-state => and

0>dtd φ 01

≥−ph

BnτIf we neglect Rsp and noting =>

Therefore, n must exceed a threshold value nth in order for Φ to increase. In other words J needs to exceed Jth in steady-state condition, when the number of photons Φ=0.

sp

thth nqdJ

τ=


⎪⎪⎩

⎪⎪⎨

⎧

=+−Φ−

=Φ

−+Φ

0

0

qdJn

BN

RBN

sp

thsth

ph

sspsth

τ

τ

This expression defines the current required to sustain an excess electron density in the laser when spontaneous emission is the only decay mechanism.

Now, under steady-state condition at the lasing threshold:

Фs is the steady-state photon density.

0=+−Φ

−qdJnR

sp

th

ph

ssp ττ

phphsp

thphsps qd

JnR τττ

τ +−=Φ

qdJn

sp

th =τ

( ) phspthph

s RJJqd

ττ

+−=Φ

# of photon resulting from stimulated emission

Adding these two equations: but

The power from the first term is generally concentrated in one or few modes;The second term generates many modes, in order of 108 modes.


To find the optical power P0:

cnLt =Δ time for photons to cross cavity length L.

sΦ21 - is the part travels to right or left (toward output

face)

R is part of the photons reflected and 1-R part will escape the facet

( )( )( )

t

RhvvolumeP

s

Δ

−⎟⎠⎞

⎜⎝⎛ Φ

=1

21

0

[ ]thph JJ

qnRWhc

P −⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

λτ

2)1(2

0

Laser Characteristics

P0

J

Jth

nth

P0 lasing output power ≡ Фs

Thresholdpopulationinversion

n

Laser CharacteristicsResonant Frequency

mL παβ 2= λπβ /2 n=

mLn πλπ 222 =⎟

⎠⎞

⎜⎝⎛

cnLv2

=λnLm 2

=

We had:

n

Lm

2λ

=

This states that the cavity resonates (i.e. a standing wave pattern exists within it) when an integer number m of λ/2 spans the region between the mirrors. Depending on the laser structures, any number of freq. can satisfy

1&)0()2( 2 == − LjeILI β

Thus some lasers are single - & some are multi-modes. The relationship between gain & freq. can be assumed to have Gaussian form:

( )2

20

2)0()( σλλ

λ−

−= egg

where λ0 is the wavelength at the center of spectrum; σ is the spectrum width of gain & maximum g(0) is proportional to the population inversion.

Laser CharacteristicsSpacing between the modes:

mvcLnm 2

= Ln

m=

2λ

121 −=− mv

cLnm

nLmnL

mnL

mnL

m 22

122 2

2

λδλ ==+

−=

( ) 1221 =Δ=− − v

cLnvv

cLn

mm Lncv

2=Δ

λvc =2λλΔ

−=Δ=>=cv

vcv

LnLncc

22

2

2

λλλ

λ=Δ=>=

Δ

Laser CharacteristicsInternal & External Quantum Efficiency

Quantum Efficiency (QE) = # of photons generated for each EHP injected into the semiconductor junction ≡ a measure of the efficiency of the electron-to-photon conversion process.If photons are counted at the junction region, QE is called internal QE (ηint), which depends on the materials of the active junction and the neighboring regions. For GaAs ηint = 65% to 100%.If photons are counted outside the semiconductor diode QE is external QE(ηext).

Consider an optical cavity of length L , thickness W and width S. Defining a threshold gain gth as the optical gain needed to balance the total power loss, due to various losses in the cavity, and the power transmission through the mirrors .


The optical intensity due to the gain is equal to:I = Io exp(2Lgth), this will be lost due to the absorption and the reflections on both ends by

R1R2 exp(-2Lα)

So: I = Io exp(2Lgth){ R1R2 exp(-2Lα)} = Io . Therefore:

1)(221 =−αthgLeRR

where R1 and R2 are power reflection coefficients of the mirrors, α is attenuation constant.


21

1ln21

RRLgth += α

g is gain constant of the active region and is roughly proportional to current density (g = βJ). β is a constant.

( )21

1ln21)/1(

RRLJth += αβ By measuring Jth, α, L, Γ1 and Γ2 one

can calculate β (dependent upon the materials and the junction structure).

The ratio of the power radiated through mirrors to the total power generated by the semiconductor junction istotal

ra

PP

RRL

RRL≡

+21

21

1ln21

1ln21

α


Therefore ηext= ηint(Pra/Ptotal) which can be determined experimentally from PI characteristic.For a given Ia (in PI curve current at point a) the number of electrons injected into the active area/sec = Ia /q and the number of photons emitted /second = Pa /hν

qIhP

a

aext /

/ νη =

qIhP

b

bext /

/ νη =

IP

hq

hIIPPq

ba

baext Δ

Δ=

−−

=νν

η)(

)(

i.e. ηext is equal to slope of IP curve in the region of I > Ith.

thext II

Phq

−=

νηIf we choose Ib = Ith , Pb≈0 and

hν≈ Eg and hν/q = Eg /q gives voltage across the junction in volts.

Laser CharacteristicsPower Efficiency

At dc or low frequency the equivalent circuit to a LASER diode may be viewed as an ideal diode in series with rs.

Therefore the power efficiency = ηp =(optical power output)/ (dc electrical power input)

sgp rIqEI

P2)/( +

=η

extthIIq

hP ην )( −=

sg

gthextp rIqEI

qEII2)/(/)(

+

−=

ηη

Documents

ELEC 425/ 6261 Lecture Notes 2 & 3