Some Fundamentals of Wave Optics - Concordia University

Some Fundamentals of Wave Optics

Optoelectronic Devices & Communication Networks

Amplifier

Add/Drop

WDM

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Montreal

Toronto

Ottawa

Optoelectronic Devices for Communication Networks

Devices to be Studied:

» Fiber Optics

» Optical Sources: LED, LASER

» Optical Diodes

» Photodetectors

» Optical Amplifiers

» Optical Attenuators/Modulators

» Optical Isolators

» Optical Switches

» WDM

» Add/Drop Devices

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. Interaction between light and solid state materials

Some Concepts of Solid State Materials

Contents

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

Light Properties

Wave/Particle Duality Nature of Light based on principle of quantum mechanics.

Observed phenomena due to the wave nature of light and related subjects to be studied:

•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•Dispersion•Sources of Dispersions

•Polarization of Light

•Elliptical and Circular Polarization

•Birefringent Optical Devices

•Electro-Optic Effects

•Magneto-Optic Effects

The Nature of LightDiffraction and Interference

The Nature of LightPhotoelectric Effect

hhhmvKE 02

2

1

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 reflection, refraction, diffraction, interference, polarization,… used to explain the concepts of light transmission in fiber optics, WDM, add/drop/ modulators,…

hpmcE

chhE 2

Since waves behave as particles, then particles should be expected to

show wave-like properties. De Broglie hypothesized that the wavelength

of a particle having momentum p can be expressed as:

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 MaxwellEquations).

Ex

z

Direction of Propagation

By

z

x

y

k

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.

Traveling wave (sinusoidal):

or 00 .cos),( rktEtrE

]Re[),( )(

00 kztjj

x eeEtzE

z

Ex = E

osin(t–kz)

Ex

z

Propagation

E

B

k

E and B have constant phase

in this xy plane; a wavefront

E

A plane EM wave travelling along z, has the same Ex (or By) at any point in a

given 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.


The wave nature of Light y

z

k

Direction of propagation

r

O

E(r,t)r

A travelling plane EM wave along a direction k.


00 cos, krtEtrE

tconskzt tan0

kdt

dz

t

zV 2Phase velocity:

]Re[),( )(

00 kztjj

x eeEtzE

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

zzk

2Phase difference:

k

Wave fronts

r

E

k

Wave fronts(constant phase surfaces)

z

Wave fronts

PO

P

A perfect spherical waveA perfect plane wave A divergent beam

(a) (b) (c)

Examples of possible EM waves


krtr

AE cos Spherical wave

00 cos, krtEtrEPlane wave

t

EE

2

22

Maxwell’s Wave Equations

Electric field component of

EM wave:

These are the solutions of Maxwell’s equation

Optical Divergence

y

x

Wave fronts

z Beam axis

r

Intensity

(a)

(b)

(c)

2wo

O

Gaussian

2w

(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).


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

)2(

42

0w

is called beam divergence

Gaussian Beams

Refractive Index

cV 00

1

0

1

V

0 r

rv

cn

Phase velocity

Speed of light

k(medium) = nk

λ(medium)= λ/n

Isotropic and anisotropic materials?; Optically isotropic/anisotropic?

n and εr are both depend on

The frequency of light (EM wave)

If the light is traveling in dielectric medium, assuming nonmagnetic

and isotropic we can use Maxwell’s equations to solve for electric

field propagation, however we need to define a new phase velocity.

εr is the relative permittivity of the medium

+

–

kEmaxEmax

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.


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.

Ng

n

500 700 900 1100 1300 1500 1700 1900

1.44

1.45

1.46

1.47

1.48

1.49

Wavelength (nm)


g

gN

c

d

dnn

c

dk

dmediumv

)(

What is dispersion?; dispersive

medium?

2

)(n

cvk

In vacuum group velocity

is the same as phase velocity.

z

Propagation direction

E

B

k

Area A

vt

A plane EM wave travelling along k crosses an area A at right angles to the

direction of propagation. In time t, the energy in the cylindrical volume Avt(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.

n

y

n2 n

1

Cladding

Core z

y

r

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, , z to represent any point in the fiber. Cladding isnormally much thicker than shown.


Structure of Fiber Optics

n2

n2

z2a

y

A

1

2 1

B

A

B

C

k1

Ex

n1

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.


n2

z

ay

A

1

2

A

C

kE

x

y

ay

Guide center

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


n2

Light

n2

n1

y

E(y)

E(y,z,t) = E(y)cos(t – 0z)

m = 0

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 . It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.


ztyEtzyE mm cos)(2,,1

mmmmm ykztykEtzyE

2

1cos

2

1cos2,, 01

y

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

Cladding

Cladding

Core 2an

1

n2

n2

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.


Low order modeHigh order mode

Cladding

Core

Ligh t pulse

t0 t

Spread,

Broadened

light pulse

IntensityIntensity

Axial

Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes whic h 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.

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

n2

z

y

O

i

n1

Ai

ri

Incident Light BiAr

Br

t t

t

Refracted Light

Reflected Light

kt

At

Bt

BA

B

A

A

r

ki

kr

A light wave travelling in a medium with a greater refractive index ( n1 > n2) suffers

reflection and refraction at the boundary.


Snell’s Law and Total Internal Reflection (TIR)

1

2

sin

sin

n

n

V

V

t

i

t

i

1

1

sin

sin

n

n

V

V

r

i

r

i

Vi = Vr , therefore θi = θr

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

1

2sinn

nc , We have total internal reflection (TIR)

n2

i

n1 > n

2

i

Incident

light

t

Transmitted

(refract ed) light

Reflected

light

kt

i>

c

c

TIR

c

Evanescent wave

ki

kr

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending onthe incidence angle with respect to c, which is determined by the ratio of the refractive

indices , the wave may be transmitted (refracted) or reflected. (a) i < c (b) i = c (c) i

> c and total internal reflection (TIR).


00

1

r

v r

v

cn

Isotropic and

anisotropic materials

zktjEezyxE izi

y

t

exp,, 0,

2

2

1

2

2

2

122 1sin

2

i

n

nn

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

x

y

z

Ey

Ex

yEy

^

xEx

^

(a) (b) (c)

E

Plane of polarization

x^

y^

E

(a) A linearly polarized wave has its electric field oscillations defined along a lineperpendicular to the direction of propagation, z. The field vector E and z define a plane o fpolariza tion. (b) The E-field os cillations are contained in the p lane of polarization. (c) Alinearly polarized light at any instant can be represented by the superposition of two fields Ex

and Ey with the right magnitude and phase.

E


k i

n2

n1 > n 2

t=90°Evanescent wave

Reflected

waveIncident

wave

i r

Er,//

Er,

Ei,

Ei,//

Et,

(b) i > c then the incident wave

suffers total internal reflection.However, there is an evanescentwave at the surface of the medium.

z

y

x into paperi r

Incident

wave

t

T ransmitt ed wave

Ei,//

Ei,Er,//

Et,

Et,

Er,

Reflected

wave

k t

k r

Light wave travelling in a more dense medium s trikes 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) i < c then some of the wave

is transmitted into the less densemedium. Some of the wave isreflected.

Ei,


Transverse electric field (TE)

Transverse magnetic Field (TM)

).(

0

rktj

iiieEE

).(

0

rktj

rrreEE

).(

0

rktj

ttteEE

zktjEezyxE izi

y

t

exp,, 0,

2

Fresnel’s Equations:

using Snell’s law, and applying boundary conditions:

21

22

2

122

,0

,0

sincos

sincos

ii

ii

i

r

n

n

E

Er

21

22,0

,0

sincos

cos2

ii

i

i

t

nE

Et

ii

ii

i

r

nn

nn

E

Er

cossin

cossin

22

122

22

122

//,0

//,0

//

2

1222//,0

//,0

//

sincos

cos2

ii

i

i

t

nn

n

E

Et

n = n2/n1

Internal reflection: (a) Magnitude of the reflection coefficients r// and rvs. angle of incidence i for n1 = 1.44 and n2 = 1.00. The critical angle is

44°. (b) The corresponding phase changes // and vs. incidence angle.

//

(b)

60

120

180

Incidence angle, i

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

| r// |

| r |

c

p

Incidence angle, i

(a)

Magnitude of reflection coefficients Phase changes in degrees

0 10 20 30 40 50 60 70 80 90

c

p

TIR

0

60

20

80


Polarization angle

i

i n

cos

sin

2

1tan

2

122

i

i

n

n

cos

sin

2

1

2

1tan

2

2

122

//

and

r‖ = r┴ = (n1 – n2)/(n1+ n2)

For incident angle close to zero:

The reflection coefficients r// and r vs. angle

of incidence i for n1 = 1.00 and n2 = 1.44.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

p r//

r

Incidence angle, i

External reflection


2

02

1or EVI

2

2

,0

2

,0

rE

ER

i

r2

//2

//,0

2

//,0

// rE

ER

i

r

2

21

21//

nn

nnRRR

2

1

2

2

,0

2

,02

t

n

n

E

EnT

i

t2

//

1

2

2

//,0

2

//,02

// tn

n

E

EnT

i

t

2

21

21//

4

nn

nnTTT

Light intensity

Reflectance

for normal incident

Transmittance

d

Semiconductor ofphotovoltaic device

Antireflectioncoating

Surface

Illustration of how an antireflection coating reduces thereflected light intensity

n1 n2 n3

AB


n1 n2

AB

n1 n2

C

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

Reflectance

(nm)

330 550 770

1 2 21

o

1/4 2/4

© 1999 S.O. Kasap, Optoelectronics (Prent ice Hall)

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Some Fundamentals of Wave Optics - Concordia University