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Optical Communication Physics
Satyam Ladva
•

The basic Communication System consists of a transmitter, reciever and a medium (channel) for the information to be sent through
...
g
...


•

Price, Security and Local Terrain eect Channel type
...


•

Wave division and Frequency division multiplexing due to Fibre optic bundles
...
Analogue Signal: Digital are more robust against noise resulting in sharper reception and
HD, can be compressed e
...
MPEG which allows bandwidth utilisation and stored on large hard drives
...


The generic, statistical physical derivation of this noise is called the uctuation-dissipation

theorem, where generalized impedance or generalized susceptibility is used to characterize the medium
...
The term
also applies to photon counting in optical devices, where shot noise is associated with the particle nature of
light
...

0

where

2
σn = n


...
A

low SNR results in binary digit errors and info
...


•

Bit Error Rate (BER) - Error rate per bit of information sent e
...


10−9 for

telephones is the maximum BER

limit
...
Twisted wire pairs regenerated every ~2km whereas optical bres
are regenerated every >100km
...
e
...


Copper wires carry voltages/current but have large attenuations at high frequencies since the signal exists
only in the skin of the wire i
...
skin depth
...


•

Co-axial cables have wide bandwidth due to using shielded, thicker wires which reduce capacitive losses
...
This is why Governments control free
space bandwidth available for radios and TV
...


•

10Hz − 20kHz
300Hz − 3
...


Bandwidth type is information dependent i
...
Human hearing range
lters, Analog radio broadcasts can reduce bandwidth to

but, by using electronic

•

Squeezing telephone calls into smaller bandwidths = more calls along channels
...


•

By sampling at the Nyquist limit, Analog signal converted into Binary code via an ADC
...


•

ADC resolution determines the number of binary digits (1 or 0) required to describe each sampled analogue
voltage
...
Not the same thing as Binary digit
...
e
...
e
...

Channel

Bandwidth (MHz)

Attenuation (dB/km)

Repeater Spacing (km)

Open Wire (O/H Cable)

0-0
...
03@1 kHz

40

Twisted Wire Pairs

0-1
...
70@1 kHz

2

Co-axial Cables

0-500

7
...
30

>100

Optical Fibre

~25

× 106

3

Coding
•

The limits to rate of information that can be carried by a channel is dependent on SNR and Channel Bandwidth
...
g
...
The redundancy protects against symbol errors in the
presence of binary digit errors
...


•

For faster techniques, advanced symbols compressed into a combination of symbol ones e
...
Morse code which
uses dots and dashes instead of letters
...


Cryptography
•

Current safe methods involve preventing the computer from nding the two prime numbers, safe numbers, in
a really large prime number
...


•

Quantum Computers is an alternative by using photon properties of: it can't be split or cloned, phase and
polarisation e
...
BB84 Single Photon protocol
...
e
...
Radio transmission
...
Therefore a carrier

wave is required which modulates the signal and, when received, is demodulated
...


In Optical bres, carrier wave is optical with frequencies

∼ 1014 Hz
...


Networks
•

A Public switching telephone network (PSTN) is where info is transmitted from point to point
...


•

A network is inecient unless many information is handled simultaneously and only possible if high bandwidth
channels e
...
optical bre, used
...


PacketSwitching:

Information sent in bundles between network nodes and the route is not always the same e
...


Internet
...


•

WAN and LAN (Think Laptop)
...


•

Packet switching used in telephones i
...
Voice over internet protocol (VOIP)
...
Bluetooth and wireless signals controlled by piconets

Piconets:

Ad-hoc Computer network linking wireless user group of devices using bluetooth technology
...
Identiying the squares as individual numbers and then dening the probability of
Choosing a square on a regular chessboard has the individual probability of

acquiring the square as

I1

or identifying the row and column which the square is where the probability of

nding the square is the addition probability of the row and column
obtained from both sets equates

row
column
P (I1 ) = P (I2 )
...


I1 = I2

row
column
I2 = I2 + I2

...


The base of this log equals

2,

represent binary digits
...


•

Using the above example, the information transfer is given by

I = log2 a posteriori prob
...

a priori prob
...
In Statistical mechanics, it is a measure
of the spreading of particles across a range (in energy) of allowed states according to probability
...
e
...
This will result in each
source having its own separate probability i
...


•

The rst symbol to appear in a long sequence of symbols transmitted in

log2 P1 bits
•

P1 , P2 , P3 ,
...

n × P1

which will result in

n × P1 ×

of information being sent
...


•

This entropy is not the same as that in thermodynamics however, the idea behind it is similar
...
e
...
They won't gain any new information
...
7

bits/letter assuming that

the probability of each letter occuring is the same
...


Redundancy
•

It describes the presence of more symbols (or information) in a message (code) than is necessary to describe
that message fully
...
During
1 → 0 or vice versa
...


An example is a language consisting of A and B only
...
100%

•

etc
...


5

•

P (T, H) = P (T ) × P (H) = 0
...
P (T, H) = P (T ) + P (H)
...


•

An evaluation of the conditional probabilities of the two-letter combinations leads to a conditional entropy of
the form

H(j|i) = −

P (i, j)logP (j|i)
...


BitErrorRate:

The lower the BER, the better the quality of the communication link
...
This eects the channel capacity, the amount of information correctly transmitted
...
These uctuations cause the level of electrons transmitted to oscillate
about an average
...

2

Johnson noise eects scale with temperature up to high frequencies, described by thermal velocity

4kB T BR

2
< vT >=

where this equation was deduced experimentally by Johnson but proved by Nyquist, theoretically,

using thermodynamics
...


•

R=

I

is current and

V

are the voltages at the nodes
...
The Voltage/Current, as a function of time, follow a straight line unless noise is present
...


The straight line is the ideal case and represents white

noise
...


PdBm = 10log( kB T B )
1mW

•

In dBm, Johnson's noise expressed as

•

An increasing BW results in frequency component being measured increasing
...


•

By taking a sample of a region in a P-N junction, electrons are seen entering randomly
...


•

It is present as a dark current where the receiver is a photodiode, even without any incident optical signal
...


i2 = 2eBId
s

resulting in high quality detectors being used in optics only
...


•

a posteriori is the probability of the message after it is received

•

If noise is neglected, the entropy

H(T x) = −

≈ 1
...
In addition

H(Rx) = −

P (yj )log(yj ) where the a priori = P (xi ) and the a posteriori
j

=

•

P (xi |yj )
...


symbols in the entire system
...
Using Bayes rule,

This new expression can be rearranged, using Bayes rule, and expanded to give

P (x, y)log2 PP (x, y)
...
H(x) is the entropy of the source whilst H(x|y) is the equivocation,
x y
which represents the additional information that would have to be supplied to correct all errors of noise
...


The venn diagram describes the above relationships if there was noise in the system
...
If
the two were dependent, but noise free, they would overlap perfectly i
...
The information at the source and
receiver is unchanged
...

rapidly varying signal with a sequence that has a time slot
sequence is

2τ

and

B =

1
2τ
...


For a binary signal, the most

allocated to each binary digit, then period of

can't be reduced, then info
...


•

Source coding aims to transfer information as quickly as possible
...
e
...
Morse code
...
e
...


Types of Source Coding
•

Equal length codes have the advantage of not needing additional space i
...


additional space symbols not

needed
...
e
...
They separate the
words easily but more suitable for small alphabets and would have very low entropy, since binary digits are
not equiprobable
...


•

Maximum entropy

p(m) =
•

1
M and

Hmax for an alphabet with M symbols depends on prob
...

H
Hmax = log2 M )
...


If symbols coded as binary words, alternative interpretation of
average code length

ηcode =
•

H
L

L=

M
m=1

P (m)lm

ηcode

(if equiprobably,

compares the dierent codes by their

where lm is the length of each codeword
...


Human tree diagram:




Symbols ordered in descending probability
Add the two symbols with the smallest probabilities and assign 1 or 0 to them (The order doesn't matter
as long as consistency is maintained)
...

From end to start, there are many dierent paths
...
Combining these results in
a codeword
...


•

Human codes are compact codes, since they assign the simplest code for the most common letters thereby
reducing information required
...


Shannon's rst theorem
•

If symbols coded in groups of length n, instead of individually, then entropy per symbol tends to source
entropy as

n → ∞
...


•

This method allows for data compression, especially for large amounts of data
...


Channel Coding
•

•

If 2 symbols A = 0 and B = 1 then
B = 111, then one binary error will
H
as R = 1 − actual
...
This increases redundancy, expressed

The main problem is increasing redundancy reduces information transfer rate
...
e
...


• (P + P )3 = P (0) + P (1) + P (2) + P (3) where P is the binary digit error rate (averaged over the whole
sample), P (0) is the probability of the information containing no errors, P (1) for a single binary error etc
...
If two more symbols were added to A and B, then (P + P ) would be the
distribution used
...


n → ∞
...


Single Parity Check Code
•

Sending the same piece of data multiple times, they can be compared and errors can be spotted via the
majority rule
...


•

Arrange data in blocks to identify single errors by adding redundancy in the form of parity bits
...
This xed length code prevents the need for a dictionary
...


•

This is only useful for single errors and information must be resent, thereby losing eciency, if multiple errors
are detected
...


10

•

Arranging codewords in block codes can identify errors easily
...


Hamming Codes
•

Extension of the block code idea to include parity checks interspersed within the data
...


•

Store data bits in 4 numbers and parity bits in another 3
...

Designate the data digits and only include 1's
...
Add parity bits with 4 data bits and you get code
...
Then XOR the transmitted parity
data with the new parity data to identify data bit in error
...


Minimum Hamming distance
•

The number of binary digits which dierentiate pairs of codewords is the Hamming distance
...


dmin :

1-5, applied to

A = 00000, B = 00111

etc
...


•

Uaing exclusive or arithmetic any two code words always generates another member of the group - closure
...


Other Codes
•

Convolution codes employ shift registers, due to the data stream being continuously fed into the encoder,
which act like memmory slots for bits and insert parity bits as the data streams pass out of the shift register
...


•

Burst errors can be dealt with by dealing with the symbols instead of the individual binary digits and
interleaving symbols to break down the eects of burst errors
...


11

Run-length coding
•

Rather than coding each inkblot 1, if there is a string of inkblots, they can be grouped and given a grouped
pixel
...


Bitmaps
•

Pointillism - An image is formed from points (pixels) of paint
...


•

Digital media e
...
inkjet printer, assign colour/intensity information to a 2D array of pixels to form a bitmap
...
Depending on the bits i
...
7-bit set, the shades of gray are
controlled
...


•

TIFF (Tagged image le format) is owned by ADOBE
...


•

If 7 binary digit code required per letter in ASCII, then 28 binary digits for 4 letter word
...


PKZIP compression
•

Utilises Human coding to produce an archive le and then applies LZW to achieve compression factors of
20 for text based les
...
e
...


MPEG
•

Analogue tv consists of screen 4:3 ratio
...
In interlaced mode, every second line scanned and
beam returns to top to ll in remaining lines
...
5MHz allocated bandwidth
...


0
...
11B )

and Chrominance (CR

Colour images sent via Luminance (Y

= 0
...
49(B − Y )

= 0
...
) with relation to Red, Green

and Blue
...
Human eye more
sensitive to changes in luminance
...


JPEG
•

Used to compress images
...


•

Due to eyes sensitivity, can average

CR

and

CB

values over adjacent rows
...


•

Each set of

Y, CR , CB

is subjected to a 2D discrete cosine transform, which calculates frequency content of a

spatially varying object
...


•

Music sections are pre-ltered into frequency bands: 0-5
...
5-11kHz and 11-22kHz then subjected to a
DCT analyses
...
This is possible by using a
masking curve
...


2
T

´

T
2

•

Multiply by

•

For purely even and odd function, one of the two constants are 0
...


•

Summing harmonics will eventually result in a square wave, with higher harmonics having reduced eect
...


a2 + b2
n
n

Cn =

an =

and

−T
2

f (t)cosnω0 t dt
...


by considering the power

Exponential form
•

Applying Euler's equations
form of the FS:

f (t) =

1
cosωt = 2 (ejωt + e−jωt )

∞
−∞

Cn ejωn t

where

ωn =

and

sinωt =

2πn
T and

Cn =

1
jωt
− e−jωt ) results in the exponential
2j (e
´T
1
−jωn t
2
dt
...


A train of square pulses
•

Amplitude

•

ωn τ
Aτ
2π
T sinc( 2 ) where the Cn are equally spaced by ωn = T with
amplitudes dictated by the sinc function
...


A,

width

τ

and period

Using the exponential form gives

T

where

Cn =

T > τ
...
All of the Cn should be positive to make the sinc function
The rst crossing point occurs at

values positive and when the pulses become narrow, the width of the sinc function broadens in the limit

τ → 0
...


constant
...


When

f (t)
...

2

If the pulse becomes innitely narrow (δ -function) the BW becomes innitely wide
...


•

delta function denition:

t0 )f (t)dt = f (t0 )
...
At

same situation but with a phase shift by a factor of

•

whereby the impulse function is dened by:

´∞
−∞

δ(t −

t = t0

the

t0
...
e
...

Perform same FT for

cosω0 t

ω = ω0

and then use the double solid frequency representation
...


Signal Filtering
•

Reduces frequency content of a signal or extracts the information from a modulated carrier wave
...
At
Vout
Vin = H(jω)
...
e
...

voltage Y (jω) =Input voltage X(jω) × H(jω)
...
g
...
The j indicates a phase shift between Vin and Vout and
Vout
ω
whilst the phase angle tanφ =
ω0
...
logω plot and a φ vs
...


Vout =

In the time domain

h(t)

is the impulse response
...
The shaded area indicates a weighted impulse signal
...
The expected response of an impulse
´∞
is δ(τ ) → h(t) which allows output voltage to be y(t) =
x(τ )h(t − τ )dτ
...


• 1 − e−t/RC =

´∞
−∞

x(τ )h(t − τ )dτ =

´t
0

1
...


This enables

h(t)

to be found which can be checked by

FT
...


• fc (t) = Acos(ωc t + φ)

where the signal can be FM, AM or PM
...

Convolving the delta functions for the carrier wave
produces a total of 4 signals
...


This

multiplication of carrier wave produces

an upper and lower sideband
...


using non-linear circuits e
...
diode, transistors
...


Vin

indicates a reverse

15

•

2
3
Vout = A1 Vin + A2 Vin + A3 Vin +
...


Fitting a polynomial of the form

Vout

where

Vin = (fs (t) + cosωc t)RL

gives

Demodulation
Homodyne detection
•

Modulated input signal (heterodyne signal) multiplied by

•

It is then passed through a LPF
...


•

Can send signal via phase locking but the technology required is very expensive
...


is convolved with the local oscillator signal
...

negative values, AC (1 + m) → AC (1 − m)
...
To prevent

whereby the method is called double sideband large carrer
...


To receive the baseband signal, time domain signal passed through a diode
...
Modulated wave smoother depending on RC circuit choice,
after diode
...
Eciency of transmission µ is ratio of sideband power to total power since

Average power = |amplitude|
...


only sideband power recoverable
...


•

DSB-LC requires twice the baseband bandwidth and is wasteful
...


Frequency Modulation
•

Shannon's information theorem states

C = Blog2 (1 +

S
reduce
N
...


•

S
N ) where

Signal wave used to modulate carrier wave of FM radio

B

is channel bandwidth
...


B

becomes, can

Can't use FT to convert

into frequency domain but requires Bessel functions
...


•

Amplitude shift keying (ASK)/On-o kying (OOK)
...


•

Modulation can be achieved by direct OOK to the drive current of the laser or operating the laser as a
continuous wave and use an external modulator
...
For optical systems, if the photodectors are fast enough, the photogenerated
current will follow the binary signal
...


•

Many reasons to convert analogue signal to digital signal:





Integrity - Digital signals are more robust against noise, attenuation and distortion
...

Technology - digital technology cheaper and more sophisticated than analogue, allowing for faster switching time
...

Security - Easy to encrypt the information
...
The wires
connecting the circuit will have an inductance/capacitance which will limit the response time to any change
...


17

Sampling Rate
•

Analogue signal sampled at a rate fast enough so no information lost and then each sample voltage allocated
a codeword
...


•

A simple sampling circuit (draw diagram) samples the analogue signal every T seconds whereby the switch
rotates with a period of T seconds
...
T is dependent of
channel response time and sampling rate
...
e
of a train of pulses, period T
...


f (ν) ⊗ s(ν) = Fs (ν)

where

provides information about the sampling rate
...


Separating the replicas

fs =

ν

represents the frequency domain and the convolution

consists of many replicas of the orignal baseband signal

1
T gives the Nyquist limit

fs < 2fm
...


How much information?
•

Range of measurable voltages depends on the degree of amplication of the thermocouple voltage
...


•

Can split the signal into sections e
...
Split a signal into 8 sections is only possible with

23

i
...
3 bit ADC
...
The limitation of the number of sections is noise i
...

Too small, and noise will be recorded
...

•

2
N + Average signal power S = Receiver power PT
...
Maximum number of level given by 2 = VN where
Average noise power

S
N ) - Amount of info obtained per sample
...
e
...


m = log2

•

(1 +

The dynamic range

D

of the ADC gives the power ratio of the maximum possible rms voltage

maximum detectable signal

Vnoise

which is measured in dB
...

The quantised signal

fq (t) ≈ f (t)
...
This variation lies between
separtion of the

•

If all values of

´

∆
2
−∆
2

M

(Mj −

∆
2)

<

< (Mj +

∆
2 ) where

∆=

2

´∞

2V
M is the

voltage levels
...
Recalling M = 2m
2
(∆ )
12
6n
...
8 +
6dB for

S
Nq
...


Non-linear Quantisation - Companding
•

In telephone systems, non-linear quantisation is used to benet quiet speakers and provide a balance between
loud and quiet speaker

•

S
Nq
...


18

•

At the transmitter, the output is compressed relative to the input according to the A-law:

Vout =

1+lnAVin
1+lnA

where A describes the curvature
...
As signal drops, quantisation becomes more severe
...


a random noise signal is added to the analogue signal before quantisation
...


How is ADC performed?
•

A clock provides the stream of pulses which are input to the binary counter
...


•

The ramp voltage is directed to the inverting input of an op-amp whilst the sampled voltage

f (t)

is fed into

the non-inverting terminal
...


•

When

f (t) =

D
A the output comparator voltage switches from positive to negative and the clock and counter

are reset
...


Specications
•

To ensure high delity reproduction, CD was designed to contain entire range of human hearing - bandwidth
of

fm

and

D > 90dB
...


•

The LPF is used to prevent aliasing by rejecting frequencies about 22kHz and dithering and sampling at
44
...


Coding
•

Digitised data encoded and redundancy added to detect and correct errors - cross interleaved reed solomon
code (CIRC)
...
These occur when areas of the disc
have been corrupted i
...
scratched, dusty etc
...


•

Symbols are cross-interleaved to prevent this hissing/jumping and Reed-Solomon block coding technique used
to identify and correct errors at the byte and not the binary bit level
...
24byte increased to 33
...


•

17 bits added to allow for synchronisation to increase total frame size to 588 bits (1/3 are data bits)
...
5Gb of storage on 60minute C with 4
...
16MHz analogue BW equivalent
...
e
...
The power is constant throughout the large
frequency range and can be ltered out by using a MHz LPF
...
A DAC is then used followed by a 20kHz LPF to reject components not within the original signal
...

The form a spiral from the centre of the disc to outer
...


•

Diraction limited spot size is dependent on laser wavelength (Semiconductor lasers) and is about

1µm

for

the lens systems in CDs
...
e
...


•

Binary 1 digits read at edge of pit and the transition from pit to land can't occur more frequently than 3 bits
i
...
merging bits
...


•

The polariser polarises the light from the laser, which is collimated
...
Reection from
the disc surface induces another phase change whilst at the prism, this light is reected towards the sensor
(not the laser)
...


Focus
•

The better the focus, the smaller the laser spot size
...


•

If the beam is focused, op-amp output voltage will be very low (∝

(S1 − S2 ))
...


Tracking
•

Inserting a diraction grating between the laser and the polarising prism generates 3 beams where each beam
reected from the disc is directed to a sensor and a comparison of the outputs maintains tracking of the central
beam
...


DVDs
•

Even with MPEG compression, digital BW high for compact discs
...


•

Changing the focus of the laser also allows for double sided discs with twice as much storage capacity
...
r−ωt)

2
2
2
k 2 = kx + ky + kz

•

Incidence Electric eld:

•

TE - Transverse electric: Light polarised perpendicular to the plane of incidence and TM - Transverse mag-

where

netic: Light polarised parallel
...


This

gives the electric elds in the form:

 E i = Ei ei(kx x+ky y−ωt)
 E r = Er e
 E t = Et e
•

- Incident eld

i(kx x+ky y−ω t)

- Reected eld

i(kx x+ky y−ω t)

- Transmitted eld

Use Maxwell's equation

× E = − ∂B
∂t

where

≡ ik

and

∂
∂t

≡ −iω

to give the corresponding magnetic ux

densities

 B i = k × Ei
ω
 Br = k
 Bt = k

•

Note:

ω
k

- Incident Magnetic eld

E
× ωr - reected magnetic eld
E
× ω t - transmitted magnetic eld

=

ω
k

=

c
ω
n1 and k

=

c
n2

Tangential components are continuous at the boundary whilst normal components are continuous across the
boundary
...


x = 0)
...
Assume that the frequency of the wave and

its wavenumber remains the same for the reected and transmitted wave
...
It

also is true otherwise waves would pile up at the boundary
...


2
2
k 2 = kx + ky = k 2

21



For the transmitted wave, can rewrite the incident and transmitted conditions from the rst condition
to determine two expressions for

•

k0
...


These boundary conditions result in the exponential terms in the tangential components cancelling giving:

Ei + Er = Et
•

Following the same procedure for the continuity of the magnetic eld vector across the boundary, you get:

kx Ei + kx Er = kx Et
Et
Er
Ei and Ei of the signal where

kx = kcosθi = ω n1 cosθi , kx = ω n2 cosθr
...
The TM wave equations are
more complicated to derive
...


•

Condition for polarisation:

θi = 0,

TE and TM modes are equal and cannnot be distinguished
...


Reection at normal incidence
2
R = rT E
...


θi = θt ,

can calculate the reectivity

Reection at other angles
•

For arbitrary polarisations, can nd reectivity by separating the incident wave into TE and TM components
and superimposing the calculated values for each
...


Transmitted amplitudes
•

Intensity of beams given by Poynting vector, time averaged:

•

The transmittance is thus:

T =

nt cos(θt ) 2
ni cos(θi ) tT E so that



where

S = c2

0E

×B

T + R = 1
...

n1

- Important for high index semiconductors - external eciency of LEDs
...
In second region,
cos(θt ) = 1 − sin2 (θt ) = ±i sin2 (θt ) − 1 gives an x-component which is

y-component same as in rst medium:

sin(θt ) > 0
imaginary:




and, by calculating

kx = n2 k0 cos(θt )
...


•

Acceptance angle:

•

NA =

sinθA
...

ˆ

22

Lectures 12 and 13 - EM modal description
EM description of modes in bres
•

TE polarised is also known as S-polarised and E-polarised (E
P-polarised and H-polarised (E

•

⊥ plane)

whilst TM polarised is also known as

plane)
...
The points which they cross are dened as modes
...


•

(r, φ, z)
•

E + (nk0 )2 E = 0
ˆ
E(r, φ, z) = rEr + φEφ + z Ez
...

z

The rst and second terms are coupled whereas the third terms is not coupled and it corresponds to the
direction of propagation in the bre so the rst step is to solve for

Ez

and

Hz ,

then using Maxwell's

equations to nd the other terms
...


Ez = R(r)Φ(φ)Z(z) where Z = eikz z −→ Propagation along the eld
imφ
is in the z-axis and kz is the propagation vector k within the waveguide
...
e
...
m is dened as the
mode index
...

1
r
r

solved by using Bessel functions where

R

only
...
This
gives two sets of solutions for

Ez

and

Hz
...




Matching Tangential eld components at core and cladding gives:

∗ Ez (r = a− ) = Ez (r = a+ )




and so forth for all

Eφ , Hz

4×4

matrix
...


You then get a set of TE modes and TM modes, the latter are the only results of the

1−D slab waveguide

mode
...

For

m = 0,

TE and TM modes are combined into hybrid modes
...


23

•

TE and TM modes can be represented on a simple ray diagram whereas EH and HE are represented by a
skew ray diagram
...
The number of approximate

V = k0 a(n2 − n2 ) 2 where
1
2
4V 2
modes is
π 2 + 2 for V < 10
...


The lowest mode - the one that always exists - is

HE11
...


Lij

where modes that are very close to each other

can be considered as a single mode
...


Lecture 14 - Attenuation, Fabrication and Photonic crystal
bers
•

As a wave moves through a material, it results in losses until the amplitude of the wave - after a large distance
- becomes very small
...

˜

Consider a 1-D plane wave:

kz = nk0
˜

where

where

kz

is a complex value
...


This part

results in attenuation
...
The main increase is in

ncore
...


•

Power loss through a ber is determined by:

Pin
αl = 10log10 ( Pout )

where

α

is the loss in dB per unit length
...
This
technique can also be used to determine losses around bends - the majority of refractive loss occurs on the

Cut-back technique - Determines the loss by using two bres, of known length
...


•

Photonic Crystal bres - Preform made by stacking glass tubes and rods together and stretching them to a
bre
...




A solid core PCF guides using one material, no dopant, with air holes which reduce the eective index
surrounding the core
...




A hollow core PCF guides in a lower index
...
High power delivery possible but for shorter distances i
...
medical devices
...
e
...


•

No dispersion results in the pulses being easily dened (You can tell what is a pulse and what isn't) but severe
dispersion results in the pulse broadening to such an extent that they overlap resulting in what seems like a
continuous pulse being measured
...
or B = ∆τ (The bit-rate is equal to the inverse
of the broadening (in time) of the pulse) resulting in a second expression: B × length = const(2) where

Dispersion increases with length travelled:

const
...

•

Intermodal dispersion: Aect Multimode bres (MM)
...

This is the cause of Pulse broadening
...


This

prevents the refractive index of the bre from remaining as a simple step function
...
The latter is like travelling in a material with a lower refractive

A simple step function dispersive power is given by:
power is:

( ∆τ )GRIN ≈
L

( ∆τ )step ≈
L

index
...


 ∆τ = τ2 − τ1 =
n1 =

∆

is the relative refractive index dierence
...


25

•

To understand other forms of dispersion, the wavepacket notion of pulses is used
...
e
...


Information travels in the wave packet so

vp > c
...


f (t)

can be FT into

F (ω)

where the pulse broadening in

f (t)

leads to the pulse width

This implies that pulse shape and duration corresponds to a superposition of signals with

dierent frequencies - power spectrum of the pulse has dierent frequency components
...


 This means that n(ω) ∝ α(ω) where the latter is absorption
...

˜
˜ ˜
c
 vp = n(λ) and vg = ∂ω ∂λ = ngc
...
This broadening
is determined by:



∆τ = D × L × ∆λ

where

D

is the dispersion parameter in ps/(nm-km)
...

broadening where at

D = 0,

The value of D determines the nature of the

minimal broadening occurs
...




Waveguide dispersion
dispersion

DM

DW

arising from

nef f (λ) is another
D = DM + DW
...


•

The most obvious solution is to use bre links greater than the limiting length however, this is not a practical
solution
...


26

•

The RX-TX are the incorporated regenerative repeaters
...
The original process is achieved electronically i
...
Optical to electrical conversion, 3R performed, then re-converted
...


•

Ampliers can be used to overcome signal loss
...
e
...
This is a coherent process
...

Fibre based ampliers - Linear based operation using rare-Earth dopants in the bre
...


Erbium doped bre amplier (EDFA)
•

Silica bre is used as a host whilst rare-earth dopants are incorporated
...
The inner
most shells are unaected and transitions here are used to build up the population inversion
...
e
...





•

Pump light is introduced via one port of wavelength selective coupler (two tapered and fused bres)
...


A problem with this is that the gain is not uniform across the spectrum (C-band) and equalisers are employed
over the full spectral width
...


•

They have typical gain of 20-30dB but requires pump laser
...


•

There are three types of dispersion:



Dispersion shifted bre (DSF) - Comprises of a material dispersion term
properties) and a waveguide dispersion term
of a ber: D

W

1 −n
= − n2 (ncλ 2 ) V

d2 (V b)
where
dV 2

D

W

b =


...


From this relation:

∆τ
L

=

D
+ DW ∆λ
...




Dispersion Flattened Fibre (DFF) - The index prole of the core changes and can be selected to provide
a at overall dispersion - or zero dispersion at two wavelengths
...




Dispersion Compensated Fibre (DCF) - Add sections of bre with opposite dispersions - especially useful
for systems where the bre is already in place
...


Semi-Conductor Optoelectronic properties
Basic Ingredients
Semiconductor materials
•

Electronic devices - e
...
Transistors, diodes, integrated circuits etc
...
SiGe and
GaAs are the two most widely used electronic materials
...
g
...
are usually based on compound
semi-conductors e
...
GaAs, GaN etc
...


•

Inorganic crystalline materials follow a cubic lattice form eg
...


28

•

Optoelectronic structures are almost always produced layer by layer where the layers are grown epitaxially
on a (relatively) thick substrate
...
The main diculty of choosing the right materials is choosing one which are close enough
to deform to substrate



a0

as well as satisfying bandgap requirements
...
There is a lattice constant a0 associated with each
GaAs
˜
pairing e
...
a0
= 5
...
From this result a0Inx Ga1-x As = xa0GaAs + (1-x)a0InAs and so, choose
x such that

a0InGaAs = a0GaAs
...
(The N-energy levels from N atoms are so closely spaced in energy
that the idea of band is more useful)
...
This results in covalent bonding
...


This results in the Brillouin zone diagrams
...
The valence band, for a
bulk material, is degenerate at k=0
...


•

Ideal Semi-conductors are intrinsic (current can only ow if electrons are promoted to the conduction band)
...


P-I-N Diode
•

Optoelectronic structures based on pn or pin diodes and use heterostructures for optical/electronic connement
...
The blue region denes holes whilst the red denes electrons
...


Optical Sources
•

Absorption is when

Eν (k) → Ec (k)

whilst emission is the opposite
...
This eective mass eects the curvature of the band
...

All of these states can take part in emission
...


•

LED's are a type of pin diode and are used because they have a wide range of spectral width (250nm 5000nm), easy to fabricate
...


extraction eciency, a broad spectral response and have a long turn on time
...
e
...
The internal power of an LED follows:

•

PLED = ηc (nqe

98% is

ω
q I)
...
Though this can't be seen, it can be used for data
...


•

Gain = absorption +

1
L ln(R) where the latter is losses from the structure with

R

being the reectivity
...
Lasing action takes place in

2nef f L
where L is the cavity length, m is the mode index (1,2,3
...
The mode spacing is given by ∆λ ≈
2nef f L which is usually very
small, so the spacings are considered even
...


d
dn

2L
n
...
dn

•

=

ment in multiple direction is possible through lateral connement via etching a ridge near the active region
to clamp down on the lateral spreading of the mode
...
Distributed Bragg reector (DBR) and Distributed feedback (DFB) lasers are
used to controllably add loss into the longitudinal modes except one
...




Feedback distributed throughout the structure and takes the form of gratings
...




The single mode systems are generally used for long-distance communications albeit the information that
can be sent is limited
...
e
...


Direct Modulation
•

Non-radiative emission - electrons excite without the need for a photon release
...
After switching o the supply at

t0 ,

the rate at which the carriers are removed by the active region

(due to recombination) with a carrier decay time constant
of LED direct modulation (∼

1ns)
...


This decay time constant is the limiting speed

The spectral width also limits LED functions
...




=


•

∂N
∂t
∂N
∂t

= −N
τ

I
qV

− (Rsp + Re + Rnv ) =
where

N
τ

t

N = N0 e− τ

A laser diode removes carriers (depopulates) through a stimulated process
...
Note: The value depends on the quality of the cavity
(and the threshold modal gain for the laser) so is associated with the design
...


•

The state below the threshold intensity

Ith

is carrier lifetime limited (like an LED) whereas above this

threshold, it is possible to achieve fast direct modulation of a laser
...
One has to consider the number of carriers but also the
Np
τP where τp is the photon lifetime in the structure and Rst links the

−

= Rsp + Rst −

electron and photon
...
e
...

Under forward bias there are many carriers that are injected which constitute a temporal loss or absorption but a change in absorption is approximately equal to a change in refractive index (from the
Kramers-Kronig)
...
As n → ∆n:
current driving laser
...
This implies that the refractive index changes depending on the

33



Regardless of the size of the change, the spectral position, the linewidth, the mode spacing etc
...


The process is known as chirp and is the most important problem against high

speed direct modulation of laser diodes
...


External modulation
•

External modulation can minimise problems such as chirp, eecting direct modulation into pin heterostructure
sources
...


Note: To achieve a reverse bias, swich the

direction of the current or switch the order of the p-i-n into n-i-p
...


•

Under a reverse bias, an electric eld (which appears across the active region) tilts the band edges
...
i
...


• F = 0

implies that a eld is present in the above diagram
...


•

Phase based external modulation is based on controlling the refractive index (optical path length) at the
operating wavelength in one arm of an interferometer (Mach-Zehnder)
...


34

•

From the rst of the two diagrams above, it is dicult to achieve equal splitting
...
When a voltage is applied to it, there is
a change in refractive index created
...

The region where the voltage is applied is known as phase shifters
...



•

From Kramers Kroning:

∆n =

´ ω2 ∆α(Voltage)
π
dω represents the refractive index change
...


•

Referring to the above diagram, if the compression process continues, quantum connement eects become
dominant and the valence band/conduction band become distorted
...


•

An exciton is a bound electron and hole pair
...


Optical Detectors
•

Photodiodes are devices that convert light energy into an electrical signal
...


•

Photodiodes are based upon the reverse bias heterostructure diode model whereby the incoming photon
generates an electron-hole pair that is accelerated (by the internal eld) and collected (in the external circuit)
as a current
...


•

By replacing

•

Based on the pin diodes, the band gap of the active region (intrinsic) places the longest wavelength that can be

Vr

in the above diagram with sunlight power, a solar cell is created
...
For p or n regions, the photons absorbed will not contribute to the current eciently
and so the band gaps sets the short wavelength limit
...






No of photons =

P0
ω where

P0

is the incident power
...


λ
= e hc P0
...




In reality:

R = (1 − R)η λq (1 − exp(−αL))
hc

where the exp refers to the fraction absorbed and

R

refers to

the reection
...




Internal quantum eciency - Ability of detector to convert photons into current:

IP =

ηeP
hν

M

is the gain
...
An applied
eld across i-region allows carriers to travel at saturation velocity (fastest speed of an electron through
a crystalline structure)
...
These velocities are smaller
than drift
...
e
...
think parallel plate capacitor
...


Combining the above 3 times results in an expression for the bandwidth of a pin diode:
where

•

A

C=

B=

1
2π [( V w

sat

)+R(Cpin +Cpkg )]

is the capacitance of the wire attached to the photodiode
...
Also, the RC time constannt and capacitance must be small
...


•

The main noise associated with photodetectors are:



Shot noise: Statistically created noise, arising from the production and collection of photo-generated
electrons upon optical illumination
...

sh

Dark Current noise: Current ow that continues to ow in the absence of light
...




i2 = 2q i B
...


wires) creates a time varying volt-

age/current
...


i2 =
T

4kB T B
which arises from
R

P = i2 R
...


•

If we are dealing with more than 20 events (event = photon or electron) then the probability distribution is
Gaussian
...


•

•

•

2
2
2
σ 2 = σsh + σD + σT
...


If two probability distributions overlap, it looks like the following:

µn e−µ
where µ is the expected number of electrons, n is the number of events
n!
−9
and the minimum number of photons required for the BER to be 10
(The maximum accepted error rate
−µ
−9
for existing telecommunication signals): P (0|1) = e
= 10 so N ≈ 21 photons
...


i2
th

is included
...
For rural settlements, the twisted pairs are replaced with co-axial
cables due to the larger BW
...
6/km
...

This avoids interferance witht the DC voltage required to power the land-line handset and reduces attenuation
in the copper wires to 0
...


•

The local exchange performs the ADC
...
Telephones require SNR of 1000 (30dB) to ensure low hiss and so an ADC with
1

M = (S/N ) 2 = 32levels

or n=5 binary digits per sample required
...
A
basic digital telephone channel (DS0) operates at a rate 64kb/s (fs

× n)
...


•

Connecting all users with co-axial cables is expensive so new users are added by utilising the additional BW
oered by the co-axial cable/optical bre allowing trac to share parts of the system
...
(Insert
exchange diagram, L12 P
...
(Insert gure 12
...


38

•

Each channel is sampled (using high quality electronics and not a rotating switch as shown) and single bits/8
bit words are input into the multiplexer
...


•

The European standard multiplexes 30 voice channels and the international standards ensures the systems
are compatible
...


•

Copper to bre change has allowed the possibility of a synchronous hierachy
...


•

SONET, in the US, and SDH, in Europe, have been developed to be compatible with existing PDH and each
other
...


•

Converting all lines into optical bres is expensive and would take 300 years
...
4, P
...


•

Using Twisted copper wires, broadband replaced dial-up but there is large attenuation in remote rural areas
...


Mobile Telephone network
•

Mobile telephone system network (insert gure 12
...


This is

usually only a few miles and, as such, mobiles require less power
...


•

Voice signal digitised and modulated onto a carrier wave which communicates with the base station
...
After leaving each
cell, a new frequency is assigned to the caller to prevent phone cuts
...
The rst mobiles used Frequency division multiple access
(FDMA) to uplink info to the base station (890-915MHz) or to download (935-960MHz) with the two channels
30kHz wide, spaced by 45MHz to avoid interferance
...
e
...
e
...
bluetooth
...




•

Broadcast networks: Devices share a common connection channel
...


LAN/MAN/WAN are the geographical denitions
...
Very resistant to failure but
becomes complicated and is expensive
...
However, it requires less links
...


Appropriate for

telephone trac but only for small data amounts
...


•

Message Switching: Message board systems store complete message at each node and transmits only when
the route becomes clear
...
This works well for emails
...
Virtual circuit is where packets follow the same route but connectionless one is where there
is no dened path thus packets are routed independently
...
The main issue
is data loss through internet transmission
...

These nodes can be connected in parallel or network may involve a series of point to point links
...


•

LAN's are broadcast networks since computers share the same transmission medium
...
They have low delay time, low error rate and are easy to install
...

Each computer connected via a tap and can send information at will providing random access (RA)
...


If packets from dierent computers meet, collisions occur

and the data is corrupted therefore LANs only work for short range
...
The utilisation

ρ=



1
(1+6
...


Token ring: (Insert diagram) Empty frames are continuously circulated and the workstation lls the rst
available empty frame adding a token + destination + message data
...
Destination W/S is reset token = 0 and reads message
data
...
Tokens avoid collisions
...


TL = τ + N I
C

N is the number of stations, I is the station latency and C is the transmission capacity
...


•

They operate in the Industrial, Scientic and Medical (ISM) band which is free to use
...


•

Low rate information x high rate chip spreading code
...
g
...


•

A slave and master i
...
printer and computer
...
e
...
The scatternet extends this idea with some
PCs acting as both masters and slaves
...


•

Each frequency is divided into time slots with a small guard band to allow radio frequencies to change

---

Title: Optical Communication Physics Notes
Description: A summary of the fourth year Imperial College London Physics lecture "OCP" whereby a physical and mathematical insight into the connection between electrical physics and optical physics, with examples of optical fibres, transmitters, forward and reverse bias, op-amps etc... is given. Includes self-generated diagrams for better understanding the concepts.

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