Optical Communication Systems 2016-2017w3.ualg.pt/~jlongras/aulas/SCO_1617_02_070217.pdf · Optical...

From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 1

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Sistemas de Comunicação Ótica

Optical Communication Systems

2016-2017

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Reviews of electromagnetism:

concepts of geometrical optics and

wave optics, and quantum physics.

S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the

publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.

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Wave nature of light

Light is an electromagnetic wave

An electromagnetic wave is a traveling wave that has time-varying electric and magnetic fields that are perpendicular to each other and the direction of propagation z.

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Ex = Eo cos(ωt − kz + φο)

Ex = Electric field along x at position z at time t

k = Propagation constant = 2π/λλ = Wavelength

ω = Angular frequency = 2πυ (υ = frequency)

Eo = Amplitude of the wave

φο = Phase constant; at t = 0 and z = 0, Ex may or may not necessarily be zero

depending on the choice of origin.

(ωt − kz + φο) = φ = Phase of the wave

This is a monochromatic plane wave of infinite extent traveling in the positive z

direction.

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Optical frequencies

Typical frequencies that are involved in

optoelectronic devices are in the infrared

(including far infrared), visible, and UV, and we

generically refer to these frequencies as optical frequencies

Somewhat arbitrary range:

Roughly 1012 Hz to 1016 Hz

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Wavefront

A surface over which the phase of a wave is constant is referred to as a wavefront

A wavefront of a plane wave is a plane perpendicular to the direction of propagation

The interaction of a light wave with a nonconducting medium (conductivity = 0) uses the electric field component Ex rather than By.

Optical field refers to the electric field Ex.

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A plane EM wave traveling 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.

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The time and space evolution of a given phase φ, for example that corresponding to a maximum field is described by

φ = ωt − kz + φο = constant

During a time interval δt, this constant phase (and hence the maximum field) moves a distance δz. The phase velocity of this wave is therefore δz/δt. The phase velocityv is

υλωδδ ===

kt

zv

Phase Velocity

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The phase differencebetween two points separated by ∆z is simply k∆z

since ωt is the same for each point

If this phase difference is 0 or multiples of 2π then the two points are in phase. Thus, the phase difference ∆φ

can be expressed as k∆z or 2π∆z/λ

Phase change over a distance ∆z

φ = ωt − kz + φο

∆φ = k∆z

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Recall that cosφ = Re[exp(jφ)]

where Re refers to the real part. We then need to take the real part of any complex result at the end of calculations. Thus,

Ex(z,t) = Re[Eoexp(jφο)expj(ωt − kz)]or

Ex(z,t) = Re[Ecexpj(ωt − kz)]

where Ec = Eoexp(jφo) is a complex number that represents the amplitude of the wave and includes the constant phase information φo.

Exponential Notation7-fev-17


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Direction of propagation is indicated with a vector k, called the wave vector, whose magnitude is the propagation constant, k = 2π/λ. k is perpendicular to constant phase planes.

When the electromagnetic (EM) wave is propagating along some arbitrary direction k, then the electric field E(r ,t) at a point r on a plane perpendicular to k is

E (r ,t) = Eocos(ωt − k⋅r + φο)

If propagation is along z, k⋅r becomes kz. In general, if k has components kx, ky and kz along x, y and z, then from the definition of the dot product, k⋅r = kxx + kyy + kzz.

Wave Vector or Propagation Vector7-fev-17


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Wave Vector k

A traveling plane EM wave along a direction k

E (r,t) = Eocos(ωt − k⋅r + φο)

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Maxwell’s Wave Equation

02

2

2

2

2

2

2

2

=∂∂−

∂∂+

∂∂+

∂∂

t

E

z

E

y

E

x

Eoro µεε


A plane wave is a solution of Maxwell’s wave equation

Substitute into Maxwell’s Equation to show that this is a solution.

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Spherical Wave

)cos( krtr

AE −= ω

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Examples of possible EM waves

Optical divergence refers to the angular separation of wave vectors on a given

wavefront.

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Gaussian Beam

Wavefronts of a Gaussian light beam

The radiation emitted from a laser can be approxima ted by a Gaussian beam. Gaussian beam approximations are widely used in photonics.

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Gaussian Beam

Intensity = I(r,z) = [2P/(πw2)]exp(−2r2/w2)

θ = w/z = λ/(πwo) 2θ = Far field divergence

The intensity across the beam follows a Gaussian di stribution

Beam axis

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The Gaussian Intensity Distribution is Not Unusual

I(r) = I(0)exp(−2r2/w2)

The Gaussian intensity distribution is also used in fiber opticsThe fundamental mode in single mode fibers can be a pproximated with a

Gaussian intensity distribution across the fiber co re

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Refractive Index

When an EM wave is traveling in a dielectric

medium, the oscillating electric field polarizes the

molecules of the medium at the frequency of the

wave

The stronger is the interaction between the field

and the dipoles, the slower is the propagation of the

wave

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Maxwell’s Wave Equation in an isotropic medium

02

2

2

2

2

2

2

2

=∂∂−

∂∂+

∂∂+

∂∂

t

E

z

E

y

E

x

Eoro µεε


A plane wave is a solution of Maxwell’s wave equation

orok µεεω 1==v

The phase velocity of this plane wave in the medium is given by

The phase velocity in vacuum is

oook µεω 1==c

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The relative permittivity εr measures the ease with which themedium becomes polarized and hence it indicates the extent ofinteraction between the field and the induced dipoles.

For an EM wave traveling in a nonmagnetic dielectric medium ofrelative permittivity εr, the phase velocity v is given by

Phase Velocity and εr

oor µεε1=ν

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Phase Velocity and εr

oor µεε1=ν

r

cn ε==

vRefractive indexndefinition

Refractive Index n7-fev-17


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Low frequency (LF) relative permittivity εr(LF) and refractive index n.

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ko Free-space propagation constant (wave vector) ko 2π/λολo Free-space wavelengthk Propagation constant (vave vector) in the mediumλ Wavelength in the medium

ok

kn =

In noncrystalline materials such as glasses and liquids, the material structure is

the same in all directions and n does not depend on the direction. The refractive

index is then isotropic

Refractive Index and Propagation Constant7-fev-17


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Refractive Index and Wavelength

λmedium = λ /n

kmedium = nkIn free space

It is customary to drop the subscript o on k and λ

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Crystals, in general, have nonisotropic, or anisotropic, properties

Typically noncrystalline solidssuch as glasses and liquids, and cubic crystals are optically isotropic; they possess only one refractive index for all directions

Refractive Index and Isotropy

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Group Velocity and Group Index

There are no perfect monochromatic waves

We have to consider the way in which a group of waves differing slightly in wavelength travel along the z-direction

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When two perfectly harmonic waves of frequencies ω − δωand ω + δω and wavevectors k − δk and k + δk interfere, they generate a wave packetwhich contains an oscillating field at the mean frequency ω that is amplitude modulated by a slowly varying field of frequency δω. The maximum amplitude moves with a wavevector δk and thus with a group velocity that is given by

vg =

dωdk

Group Velocity and Group Index7-fev-17


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Group Velocity

Consider two sinusoidal waves that are

close in frequency, that is, they have

frequencies ω − δω and ω + δω.

Their wavevectors will be k − δk and k + δk.

The resultant wave is

Ex(z,t) = Eocos[(ω − δω)t − (k − δk)z] + Eocos[(ω + δω)t − (k + δk)z]

By using the trigonometric identity

cosA + cosB = 2cos[1/2(A − B)]cos[1/2(A + B)]

we arrive at

Ex(z,t) = 2Eocos[(δω)t − (δk)z][cos(ωt − kz)]

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Two slightly different wavelength waves traveling in the same direction result in a wave packet that has an amplitude variation that travels at the group velocity.

dk

dω=gv


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This represents a sinusoidal wave of frequency ω . This is amplitude modulated by a very slowly varying sinusoidal of frequency δω. This system of waves, i.e. the modulation, travels along z at a speed determined by the modulating term, cos[(δω)t − (δk)z]. The maximum in the field occurs when [(δω)t − (δk)z] = 2mπ = constant (m is an integer), which travels with a velocity

dz

dt=

δωδk

or

dk

dω=gv

This is the group velocity of the waves because it determines the speed of propagation

of the maximum electric field along z.

Ex(z,t) = 2Eocos[(δω)t − (δk)z][cos(ωt − kz)]


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The group velocity therefore defines the speed with which

energy or information is propagated.

ω = 2πc/λo and k = 2πn/λo, λo is the free space wavelength.

Differentiate the above equations in red

dω = −(2πc/λo2)dλo

oo

ooo dd

dndndk λ

λλπλλπ

+−= )/2()/1(2 2

vg =

dωdk

oo

oo dd

dnndk λ

λλλπ

−−= )/2( 2

ooo

ooo

oo

d

dnn

c

dd

dnn

dc

dk

d

λλλ

λλλπ

λλπω

−=

−−

−==)/2(

)/2(

2

2

gv∴


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where n = n(λ) is a function of the wavelength. The group velocity vg in a medium is given by,

v g (medium) =dωdk

=c

n − λ dn

dλThis can be written as

v g ( medium) =

c

N g

Group Velocity and Group Index

N g = n − λ dn

d λ

Group Indexis defined as the group index of the medium

In general, for many materials the refractive index n

and hence the group index Ng depend on the

wavelength of light. Such materials are called

dispersive


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Refractive index n and the group index Ng of pure SiO2 (silica) glass as a function of wavelength.

Refractive Index and Group Index


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Magnetic Field, Irradiance and Poynting Vector

The magnetic field (magnetic induction) component By always accompanies

Ex in an EM wave propagation.

If v is the phase velocityof an EM wave in an isotropic dielectric medium

and n is the refractive index, then

yyx Bn

cBE == v

where v = (εoεrµo)−1/2 and n = ε1/2

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Photon energyA photon is an elementary particle, the quantum of the electromagnetic field

including electromagnetic radiation such as light.

ν= hphotonΕ h is the Planck's constant (6.6×10-34 Js).

ν is frequency of the light.


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A plane EM wave traveling along k crosses an area A at right angles to the direction ofpropagation. In time ∆t, the energy in the cylindrical volume A× ∆ t (shown dashed) flowsthrough A.

EM wave carries energy along the direction of propagation k.

What is the radiation power flow per unit area?


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As the EM wave propagates in the direction of the

wavevector k, there is an energy flow in this direction. The

wave brings with it electromagnetic energy.

The energy densities in the Ex and By fields are the same,

22

2

1

2

1y

oxro BE

µεε =

The total energy density in the wave is therefore εoεrEx2.

Energy Density in an EM Wave


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If S is the EM power flow per unit area,

S = Energy flow per unit time per unit area

yxroxroxro BEE

tA

EtAS εεεεεε 22

2 ))((vv

v ==∆

∆=

In an isotropic medium, the energy flow is in the direction of wave

propagation. If we use the vectors E and B to represent the electric and

magnetic fields in the EM wave, then the EM power flow per unit area

can be written as

Poynting Vector and EM Power Flow

S = v2εoεrE×B

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where S, called the Poynting vector, represents the energy flow per unit time per unit area in a direction determined by E×B(direction of propagation). Its magnitude, power flow per unit area, is called the irradiance (instantaneous irradiance, or intensity).

The average irradianceis

221

average oro ESI εεv==

Poynting Vector and Intensity


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Since v = c/n and εr = n2 we can write

23221

average )1033.1( ooo nEnEcSI −×=== ε

The instantaneous irradiance can only be measured if the

power meter can respond more quickly than the oscillations

of the electric field. Since this is in the optical frequencies

range, all practical measurements yield the average irradiance because all detectors have a response rate much

slower than the frequency of the wave.

Average Irradiance or Intensity


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Irradiance of a Spherical Wave

24 r

PI o

π=

Perfect spherical wave

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Spherical wave front

9A4AAA

r

2r

3r

OSource

Po

Irradiance of a Spherical Wave

24 r

PI o

π=


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A Gaussian Beam

I(r, z) = Imaxexp(−2r2/w2)

I(r,z) = [2Po/(πw2)]exp(−2r2/w2)

θ = w/z = λ/(πwo) 2θ = Far field divergence

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Power in a Gaussian Beam

])/(2exp[)0()( 222 wrIrI −=

865.0

2)(

2)(

0

0 =

∫

∫∞

rdrrI

rdrrIw

π

πFraction of optical power

within 2w=

Area of a circular thin strip (annulus) with radius r is 2πrdr. Power passing through this strip is proportional to I(r) (2πr)dr

and


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221

o

oo w

PI

π=

Io = Maximum irradiance at the center r = 0 at the waist

Example on

Gaussian Beam

Example 1.4.2 Power and irradiance of a Gaussian beamConsider a 5 mW HeNe laser that is operating at 633 nm, and has a spot size

that is 1 mm. Find the maximum irradiance of the beam and the axial

(maximum) irradiance at 25 m from the laser.

SolutionThe 5 mW rating refers to the total optical power Po available, and 633 nm is

the free space output wavelength λ. Apply

)( 221

ooo wIP π=

]m)105.0([W105 23213 −− ×=× πoI

Io = 1.273 W cm−2

∴

Corrected

7-Feb-2017

Optical Communication Systems 2016-2017w3.ualg.pt/~jlongras/aulas/SCO_1617_02_070217.pdf · Optical...

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