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Optical Devices for High Speed Communication Systems

Lecture Notes

Optoelectronic Devices & Communication Networks

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Optoelectronic Devices for Communication Networks

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Optoelectronic Devices for Communication Networks

• Requirements to understand the concepts of Optoelectronic Devices:

1. We need to study concepts of light properties

2. Some concepts of solid state materials in particular semiconductors.

3. Light + Solid State Materials

Light Properties

• Wave/Particle Duality Nature of Light

• Reflection• Snell’s Law and Total Internal Reflection (TIR)• Reflection & Transmission Coefficients• Fresnel’s Equations• Intensity, Reflectance and Transmittance

• Refraction• - Refractive Index

• Interference• - Multiple Interference and Optical Resonators

• Diffraction• Fraunhofer Diffraction• Diffraction Grating

Light Properties

• Dispersion

• Polarization of Light• Elliptical and Circular Polarization• Birefringent Optical Devices• Electro-Optic Effects• Magneto-Optic Effects

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 nature of light

Wave/Particle Duality Nature of Light

--Particle nature of light (photon) is used to explain the concepts of solid state optical sources (LASER, LED), optical detectors, amplifiers,…--The wave nature of light is used to explain refraction, diffraction, polarization,… used to explain the concepts of light transmission in fiber optics, WDM, add/drop/ modulators,…

λλν hpmcEchhE ==== 2

The wave nature of Light

• Polarization• Reflection• Refraction• Diffraction• Interference

To explain these concepts light can be treated as rays (geometrical optics) or as an electromagnetic wave (wave optics, studies related to Maxwell Equations).

An electromagnetic wave consist of two fields:• Electric Field• Magnetic Field

Direction of Propagation

An electromagnetic wave is a travelling wave which has timevarying electric and magnetic fields which are perpendicular to eachother and the direction of propagation, z.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

An electromagnetic wave consist of two components; electrical field and magnetic field components.

k is the wave vector, and its magnitude is 2π/λ

Light can treated as an EM wave, Ex and By are propagating through space in such a way that they are always perpendicular to each other and to the direction of propagation z.

We can treat light as an EM wave with time varying electric and magnetic fields.

Ex and BY which are propagating through space in such a way that they are always perpendicular to each other and to the direction of propagation z.

( )00 .cos),( φω +−= rktEtrE

Ex = Eosin(ωt–kz)Ex

Propagation

E and B have constant phasein this xy plane; a wavefront

A plane EM wave travelling along z, has the same Ex (or By) at any point in agiven xy plane. All electric field vectors in a given xy plane are therefore in phase.The xy planes are of infinite extent in the x and y directions.

( ) ( )00 cos, φω +−= krtEtrE

tconskzt tan0 =+−= φωφ

νλω===

dztzV πνω 2=Phase velocity

]Re[),( )(0

0 kztjjx eeEtzE −= ωφ

During a time interval Δt, a constant phase moves a distance Δz,

λπφ zzk Δ

=Δ=Δ2

kDirection of propagation

E(r,t)r′

Wave fronts

Wave fronts(constant phase surfaces)

Wave fronts

A perfect spherical waveA perfect plane wave A divergent beam

(a) (b) (c)

Examples of possible EM waves

( )krtrAE −= ωcos Spherical wave

( ) ( )00 cos, φω +−= krtEtrE Plane wave

∂∂

=∇ εμ Maxwell’s Wave Equations

Electric field component of EM wave:

These are the solutions of Maxwell’s equation

Wave fronts

z Beam axis

Intensity

Gaussian

(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam crosssection. (c) Light irradiance (intensity) vs. radial distance r from beam axis (z).© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Wo is called waist radius and 2Wo is called spot size

0wπλθ = Is called beam divergence

Refractive Index

cV ==00

0εεε r=

Phase velocity

Speed of light

rvcn ε== k(medium) = nk

λ(medium)= λ/n

Isotropic and anisotropic materials?

ω + δω

ω – δω

δkEmax Emax

Wave packet

Two slightly different wavelength waves travelling in the samedirection result in a wave packet that has an amplitude variationwhich travels at the group velocity.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

dz/dt = δω/δk or Vg = dω/dk group velocity

Refractive index n and the group index Ng of pureSiO2 (silica) glass as a function of wavelength.

500 700 900 1100 1300 1500 1700 19001.44

Wavelength (nm)

cdkdmediumv =

⎥⎦⎤

⎢⎣⎡⎥⎦

⎤⎢⎣

λω 2

)(ncvk

Propagation direction

Area A

A plane EM wave travelling along k crosses an area A at right angles to thedirection of propagation. In time Δt, the energy in the cylindrical volume AvΔt(shown dashed) flows through A .

In EM wave a magnetic field is always accompanying electric field, Faraday’s Law. In an isotropic dielectric medium Ex = vBy = c/n (By),where v is the phase velocity and n is index of refraction of the medium.

Cladding

Core z

Fiber axis

The step index optical fiber. The central region, the core, has greater refractiveindex than the outer region, the cladding. The fiber has cylindrical symmetry. Weuse the coordinates r, φ, z to represent any point in the fiber. Cladding isnormally much thicker than shown.

Structure of Fiber Optics

θA′

Cπ−2θ 2θ−π/2

Two arbitrary waves 1 and 2 that are initially in phase must remain in phaseafter reflections. Otherwise the two will interfere destructively and cancel eachother.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Guide center

π−2θ

Interference of waves such as 1 and 2 leads to a standing wave pattern along the y-direction which propagates along z.

E(y,z,t) = E(y)cos(ωt – β0z)

Field of evanescent wave(exponential decay)

Field of guided wave

The electric field pattern of the lowest mode traveling wave along theguide. This mode has m = 0 and the lowest θ. It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.

( ) ( )ztyEtzyE mm βω −= cos)(2,,1

( ) ⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ += mmmmm ykztykEtzyE φβωφ

21cos2,, 01

E(y)m = 0 m = 1 m = 2

Cladding

Core 2an1

The electric field patterns of the first three modes (m = 0, 1, 2)traveling wave along the guide. Notice different extents of fieldpenetration into the cladding.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Low order modeHigh order mode

Cladding

Light pulse

Spread, Δτ

Broadenedlight pulse

IntensityIntensity

Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes which then propagate at differentgroup velocities down the guide. At the end of the guide, the modes combine toconstitute the output light pulse which is broader than the input light pulse.

θrθi

Incident Light BiAr

θt θt

Refracted Light

Reflected Light

A′′θr

A light wave travelling in a medium with a greater refractive index ( n1 > n2) suffersreflection and refraction at the boundary.

Snell’s Law and Total Internal Reflection (TIR)

sinsin

i ==θθ

sinsin

i ==θθ

Vi = Vr , therefore θi = θr

When θt reaches 90 degree, θi = θc called critical angle

2sinnn

c =θ , We have total internal reflection (TIR)

n 1 > n2θi

Incidentlight

Transmitted(refracted) light

Reflectedlight

θi>θcθc

TIRθc

Evanescent wave

k i k r

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending onthe incidence angle with respect to θc, which is determined by the ratio of the refractiveindices, the wave may be transmitted (refracted) or reflected. (a) θi < θc (b) θi = θc (c) θi> θc and total internal reflection (TIR).

1μεε r

cn ε==

Isotropic and

anisotropic materials

( ) ( )zktjEezyxE iziy

t −= −⊥ ωα exp,, 0,

122 1sin2

⎥⎥⎦

⎢⎢⎣

⎡−⎟⎟

⎞⎜⎜⎝

⎛= in

α2 is the attenuation coefficient and 1/ α2 is called penetration depth

n1 > n 2

θt=90°Evanescent wave

Reflectedwave

Incidentwave

θi θr

Er,⊥Ei,⊥

Et,⊥

(b) θi > θc then the incident wavesuffers total internal reflection.However, there is an evanescentwave at the surface of the medium.

x into paper θi θr

Incidentwave

Transmitted wave

Ei,⊥ Er,//

Et,⊥

Er,⊥

Reflectedwave

Light wave travelling in a more dense medium strikes a less dense medium. The plane ofincidence is the plane of the paper and is perpendicular to the flat interface between thetwo media. The electric field is normal to the direction of propagation . It can be resolvedinto perpendicular (⊥) and parallel (//) components

(a) θi < θc then some of the waveis transmitted into the less densemedium. Some of the wave isreflected.

Ei,⊥

Transverse electric field (TE)

Transverse magnetic Field (TM)

→→−= rktj

iiieEE ω

→→−= rktj

rrreEE ω

→→−= rktj

ttteEE ω

Fresnel’s Equations:

using Snell’s law, and applying boundary conditions:

[ ][ ]2122

sincos

−−==

⊥⊥ [ ]2

sincos

−+==

⊥⊥

[ ][ ] ii

cossin

cossin22

//,0//

−−== [ ]2

1222//,0

//,0//

sincos

−+==

n = n2/n1

Internal reflection: (a) Magnitude of the reflection coefficients r// and r⊥vs. angle of incidence θi for n1 = 1.44 and n2 = 1.00. The critical angle is44°. (b) The corresponding phase changes φ// and φ⊥ vs. incidence angle.

Incidence angle, θi

00.10.20.30.40.50.60.70.80.9

0 10 20 30 40 50 60 70 80 90

| r// |

| r⊥ |

Magnitude of reflection coefficients Phase changes in degrees

0 10 20 30 40 50 60 70 80 90

−120

−180

2tannn

Polarization angle

cossin

22 −=⎟

⎠⎞

⎜⎝⎛

⊥[ ]

θπθ

cossin

21tan 2

=⎟⎠⎞

⎜⎝⎛ +and

The reflection coefficients r// and r⊥ vs. angleof incidence θi for n1 = 1.00 and n2 = 1.44.

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

0 10 20 30 40 50 60 70 80 90

θp r//

External reflection

1or EVI εε=

⊥⊥ == r

//,0// r

21// ⎟⎟

⎞⎜⎜⎝

⎛+−

=== ⊥ nnnnRRR

,02⊥

⊥⊥ ⎟⎟

⎞⎜⎜⎝

⎛== t

//,02// t

⎟⎟⎠

⎞⎜⎜⎝

( )221

=== ⊥

Light intensity

Reflectance

for normal incident

Transmittance

Semiconductor ofphotovoltaic device

Antireflectioncoating

Surface

Illustration of how an antireflection coating reduces thereflected light intensity

n1 n2 n3

Schematic illustration of the principle of the dielectric mirror with many low and highrefractive index layers and its reflectance.

Reflectance

λ (nm)0

330 550 770

1 2 21

λ1/4 λ2/4

Lightn2

A planar dielectric waveguide has a central rectangular region ofhigher refractive index n1 than the surrounding region which hasa refractive index n2. It is assumed that the waveguide isinfinitely wide and the central region is of thickness 2a. It isilluminated at one end by a monochromatic light source.

n1 > n2

Light Ligh

d = 2a

θθk1

A light ray travelling in the guide must interfere constructively with itself topropagate successfully. Otherwise destructive interference will destroy thewave.

( )πφθ

λπ man

mm =−⎥⎦⎤

⎢⎣⎡ cos22 1

mmmnk θλπ

θβ sin2sin 11 ⎟

⎠⎞

⎜⎝⎛==

mmmnkk θλπ

θ cos2cos 11 ⎟

⎠⎞

⎜⎝⎛==

ΔΦ (AC) = k1(AB + BC) - 2Φ = m(2π), k1 = kn1 = 2πn1/λ

m=0, 1, 2, …

Waveguide condition

E(y,z,t) = E(y)cos(ωt – β0z)

Field of evanescent wave(exponential decay)

Field of guided wave

The electric field pattern of the lowest mode traveling wave along theguide. This mode has m = 0 and the lowest θ. It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.

( ) ( )ztyEtzyE mm βω −= cos)(2,,1

( ) ⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ += mmmmm ykztykEtzyE φβωφ

21cos2,, 01

E(y)m = 0 m = 1 m = 2

Cladding

Core 2an1

The electric field patterns of the first three modes (m = 0, 1, 2)traveling wave along the guide. Notice different extents of fieldpenetration into the cladding.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Low order modeHigh order mode

Cladding

Light pulse

Spread, Δτ

Broadenedlight pulse

IntensityIntensity

Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes which then propagate at differentgroup velocities down the guide. At the end of the guide, the modes combine toconstitute the output light pulse which is broader than the input light pulse.

Ey (m) is the field distribution along y axis and constitute a mode of propagation.

m is called mode number. Defines the number of modes traveling along the waveguide. For every value of m we have an angle θm satisfying the waveguide condition provided to satisfy the TIR as well. Considering these condition one can show that the number of modes should satisfy:

m = ≤ (2V – Φ)/π

V is called V-number ( )21

2 nnaV −=λπ

For V ≤ π/2, m= 0, it is the lowest mode of propagation referred to single mode waveguides.

The cut-off wavelength (frequency) is a free space wavelength for v = π/2

E// Ey

(b) TM mode(a) TE mode

x (into paper)

Cladding

λ2 > λ1λ1 > λc

ω2 < ω1ω1 < ωcut-off

vg2 > vg1

The electric field of TE0 mode extends more into thecladding as the wavelength increases. As more of the fieldis carried by the cladding, the group velocity increases.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Cladding

Core z

Fiber axis

The step index optical fiber. The central region, the core, has greater refractiveindex than the outer region, the cladding. The fiber has cylindrical symmetry. Weuse the coordinates r, φ, z to represent any point in the fiber. Cladding isnormally much thicker than shown.

For the step index optical fiber Δ = (n1 – n2)/n1

is called normalized index difference

Fiber axis

Skew ray1

Fiber axis

3Meridional ray

(a) A meridionaray alwayscrosses the fibeaxis.

(b) A skew raydoes not haveto cross thefiber axis. Itzigzags aroundthe fiber axis.

Illustration of the difference between a meridional ray and a skew ray.Numbers represent reflections of the ray.

Along the fiber

Ray path projectedon to a plane normalto fiber axis

Ray path along the fiber

Cladding

The electric field distribution of the fundamental modin the transverse plane to the fiber axis z. The lightintensity is greatest at the center of the fiber. Intensitypatterns in LP01, LP11 and LP21 modes.

(a) The electric fieldof the fundamentalmode

(b) The intensity inthe fundamentalmode LP01

(c) The intensityin LP11

(d) The intensityin LP21

( ) ( )ztjrEE lmlmLP βωϕ −= exp.

LPs (linearly polarized waves) propagating along the fiber have either TE or TM type represented by the propagation of an electric field distribution Elm(r,Φ) along z.

ELP is the field of the LP mode and βlm is its propagation constant along z.

V-number

( ) ( )21

21 222

Δ=−= nnannaVλπ

( ) 405.2221

21 =−=− nnaV

coffcut λ

( ) ( ) 21

21121 2// nnnnnn −≈−=Δ

Normalized index difference

For V = 2.405, the fiber is called single mode (only the fundamental mode propagate along the fiber). For V > 2.405 the number of mode increases according to approximately

0 2 4 61 3 5V

nKb−

Cladding

Coreα < αmax

θ < θc

θ> θc

α> αmax

Propagates

Maximumacceptance angleαmax is that whichjust givestotal internal reflectionat thecore-claddinginterface, i.e.whenα=αmax thenθ=θc.Rays withα>αmax (e.g. rayB) become refractedandpenetrate the claddingandareeventuallylost.

Fiber axis

maxsinn

nn −=α

21 )( nnNA −=

0maxsin

NAaVλπ2

Optical Devices for High Speed Communication Systemsusers.encs.concordia.ca/~mojtaba/Optical...

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