1st tranche slides
TRANSCRIPT
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Outline of the III Module of the course
Modal propagation and transmission properties of dielectricwaveguides (Dielectric slab, Optical Fiber)
Optical transmitters, receivers and main optical devices:
fundamentals and principal characteristics.
Engineering and design aspects of present optical transmissionsystems.
(Course notes: http://elearning.ing.unibo.it/)
https://campus.cib.unibo.it/https://campus.cib.unibo.it/ -
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Application scenarios of optical fiber links
mono-modefiber
fiber/copper
mono-modefiber
multi-modefiber
multi-modefiber
IP network
CentralOffice
Access Networks
In-Building Networks
Core Network
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Basic Fiber Optic Communication System
DirectlyModulated
Transmitter(Based on LED
or LASER)
Input
ElectricalSignal(DigitalOrAnalog)
OpticalFibre
Direct DetectionReceiver
(Based on PIN orAPD)
OutputElectricalSignal(DigitalOrAnalog)
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Ex.: Optical Fibre Submarine systems
Up to 9000 kmof total length
National,Transoceanicand
Intercontinentallinks
Very highcapacity,high qualityrequred.
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Ex.: Optical Fibre Access Network
Typical distances1 20 km
MetropolitanAreaNetwork (MAN),Fiber to the home
(FTTH),Fiber to the Curb(FTTC)
Highcapacity,high quality
requred.
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Radio coverage of:
Crowded sites
Non-LOS Areas
(Line Of Sight)
Unit (RAU)IndoorRAU
Radio BaseStation (RBS)
Outdoor Remote Antenna
OpticalFiber
Ex.: Fiber Distributed Antenna Systems
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Advantages of Fiber Optic Transmission
High Bandwidth (
High Bit Rate )
Low Attenuation
Short Dimensions
No interference
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Band utilized in optical communications
SonarServo-
mechanisms
Radio
Radar
Infrared
Visible Light
Ultraviolet
X RaysGamma
Rays
Cellular
Telephones
TV
1 Mm 1 km 1 m 1 mm 1 nm1 mm
1 kHz 1 MHz 1 GHz 1 THz 1015 Hz 1018 Hz(f)
(l)
Frequency Band utilized in Optical Communications
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Band expressed as Dfor DlFrom ITU-T Recommendation G-692.
Nominal Frequencies in C band (1528-1561 nm):
81 carriers separated by Df=50 GHz starting from 196.1 THzIn terms ofl the range goes from 1528.77 nm to 1560.61 nm
The interval Dl is not constant. It goes from 0.389 nm to 0.405 nm
196.1 THz196.05 THz
192.1 THz
1528.77 nm 1560.61 nm
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Review on dB,dBm, dB(Hz), etc.
Attenuation : P1/P2
Attenuation|Logarithmic units = 10 log10(P1/P2) (dB)
Gain/Loss: P2/P1
Gain/Loss|Logarithmic units = 10 log10(P2/P1) (dB)etc.
P1 P2Adimensional quantities dB:
Note : AttenuationFIBER=10aL/10
AttenuationFIBER|Logarithmic units =aL (dB) [a]=[dB/km]
Power Level|Logarithmic units : dB(W), dB(mW) [or dBm], etcBandwidth|Logarithmic units : dB (Hz)
Temperature|Logarithmic units : dB (K)
etc.
Non-Adimensional quantities (W, Hz ) dB(W), dB(Hz)
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The Optical Transmission Channel
Electromagnetic Fundamentals to be recalled:
Plane waves propagation in homogeneous media
Continuity Conditions / Reflection Coefficients of the
electromagnetic field
Dielectric slab: analythical resolution
From the dielectric slab to the optical fiber
Transmission characteristics
Attenuation
Dispersion
Non linearities (not treated here)
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The Dielectric Slab: Description
x
z
(n2)
(n1)
0...
y
y
(x=d)
(x=-d) (n2)
Waveguide invariant
in the y and z directions.
Central layer
of refraction index n1 and
of thickness 2d
Two external Layersof refraction indexes n2 < n1,
infinitely extended in the
+x andx directions.
The field is invariant in the y direction
The field propagates in the z directionzjzj (x)eH(x)eE(x,z)(x,z) ,H,E
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Procedure followed to find Slab guided modes
1. Resolution ofMaxwells Equations in the three regions
Plane Wave solutions in the three regions2. Imposition of
continuity conditions of at the interfaces through either:
Coefficients GTEand GTM+ consistency condition
on the equiphase planes
Achievement of the Characteristic Equation (TE/TM, even/odd)
3. Resolution of the Characteristic Equation
Modes propagation constant
Modes amplitude behavior
Creation and Resolution of a
Homogeneous linear System
yyzzHEHE ,,,
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Plane Wave solutions in the three materials: TE case
x
z
y
zjx
yy eeEzx 222 ),(E
Upper layer:evanescent plane wave
zjxjk
y
zjxjk
yy
eeE
eeEzx
x
x
1
1
1
11 ),(
E
Central layer:
uniform plane waves
zjx
yy eeEzx 2'2'2 ),(E
Lower layer:
evanescent plane wave
1xk
1xk
ib1
rb1
22jk
x
2'2jk
x
2
02
22
'2
2
2
2
2
22
0
2)(;'2,2,1,)( knkkiknk xxixi where:
In each one of the three layers, we have zxy zxy HHHEE ,
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Reflection Coefficients at the interfaces: TE case
x
z
y
G
1
21
1
1
2
21
21
21
21
21
21
1
1
coscos
coscos
x
x
x
kjtg
x
x
xx
xx
ti
ti
djky
djk
y
TE
e
jk
jk
kk
kk
nn
nn
eE
eE
22 jkx
2'2jk
x
( )
( )
G 121
1
1 2
1
1... x
x
x
kjtg
djk
y
djk
y
TE eeE
eE
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x
z
y
The triangular points belongingto the same equi-phase plane must be
separated in phaseby an integer multiple of 2p.
The same for the square points.
Considering e. g. points A and C, itmust be:
A
'C
'
B''
B
''C
TE
CBjn
TEABjn
Ay
mj
AyCy
e
eE
eEE
G
G
'''01
'
01
''
1
2
11
p
Consistency Condition on Equiphase planes: TE case
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Dipartimento di ElettronicaInformatica e Sistemistica
G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
It must then be:
p
jm
kjtg
djn
kjtg
djn
e
e
e
e
e
x
i
x
ii
2
2
cos
2
2
2coscos
2
1
21
01
1
21
01
x
z
y
A
'C
'
B''
B
''C
i
dCB
cos
2'''
i
i
dAB
2cos
cos
2'
+d
-d
)( 'CBspan ''
)( 'ABspan
TEG
TEG
Consistency Condition on Equiphase planes: TE case
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Characteristic Equation: TE case
2244
244cos
24)2cos1(cos
2
112
1
21
1
1
21
01
1
21
01
22cos
2
22coscos
2
1
21
011
21
01
pp
p
p
p
mdktgdkdmk
tgdk
mk
tgdn
mk
tgd
n
eeeee
xx
x
x
x
i
x
i
i
jmkjtgd
jnkjtgd
jnxix
ii
( )
( ) oddmdkdkd
evenmdktgdkd
xx
xx
,
,
112
112
cot
We obtain then:
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
u
w
p/2 p 3p/2
Curve
where v is the
Normalized Frequency:
2
2
2
10
2
2
2
1 )()(
nnd
ddkv x
Odd Modes
( )( ) )(cot)cot()()()()(
112
112
modesoddeven modes
uuwdkdkdutguwdktgdkd
xx
xx
21
22 )( dkvd x
Resolution of the Characteristic Equation: TE case
From the characteristic equation we have:
Moreover, it must be: ( ) ( ) ( ) 22222212
0
2
2
2
1 )( vwunndddkx Even Modes
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G. Tartarini Advanced Electromagnetic Transmission Techniques and DevicesElectromagnetic Technologies for Link Design M
Exploiting the relations:
We can solve numerically
the equation (e.g. TE case withm=0):
For w~0 it is ~n20. For w it is~n10 .
2
02
2
2
22
011
)()(
)()(
dndd
ddndkx
])()([)()()()( 220122
012
022 ddntgddndnd
w
1001 nn mw
2002 nn mw
(w)
...
Dispersion curves (w): TE case
m=0(TE0)
m=1(TE1)
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The following relations areexpoited:
The plot is more readable
The curves can be referred to waveguides different from each other
Only one TE mode propagates when it is 0
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Behavior of |Ey|for the modes TEm (m even, m=2k)
x
z
y
G
11
2222
2
2
1
1
11
1
21
1
yy
djkkdkj
kjtg
djk
y
yTE
EE
ee
e
eEE
xx
x
x
p
Considering for example the upper interface, we have from the characteristic equation:
In the central layer the field Ey is given by:
) ( ) zjxyzjxjkxjkyy exkEeeeEzx xx 1111 cos),( 11E
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Behavior of |Ey|for the modes TEm (m even)
x
z
d-d
|Ey1|Ex. mode TE0
x
z
d-d
|Ey1|Ex. mode TE2
( ) ( )xkEexkEzx xyzj
xyy 11111 coscos),(
EIn the central layerthe expression is given by
222
0 11ppp
xkdk xx
2
3
2
3
2
311
pppp xkdk xx
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Behavior of |Ey|for the modes TEm (m odd, m=2k+1)
x
z
y
G
11
22)12(2
2
2
1
1
11
1
21
1
yy
jdjkkdkj
kjtg
djk
y
yTE
EE
eee
e
eEE
xx
x
x
pp
Considering for example the upper interface, we have from the characteristic equation:
In the central layer the field Ey is given by:
) ( ) zjxyzjxjkxjkyy exkEeeeEzx xx 1111 sin),( 11E
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Behavior of |Ey|for the modes TEm (m odd)
x
z
d
-d
|Ey1|Ex. mode TE1
x
z
d-d
|Ey1|Ex. mode TE3
( ) ( )xkEexkEzx xyzj
xyy 11111 sinsin),(
EIn the central layerthe expression is given by
pppp
xkdk xx 112
pppp 2222
3 11 xkdk xx
In general |Ey|of mode TEm assumesm times the `0` value for dxd
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Plane Wave solutions in the three materials: TM case
x
z
y
zjx
yy eeHzx 222 ),(H
Upper layer:evanescent plane wave
zjxjk
y
zjxjk
yy
eeH
eeHzx
x
x
1
1
1
11 ),(
H
Central layer:uniform plane waves
zjx
yy eeHzx 2'2'2 ),(H
Lower layer:
evanescent plane wave
1xk
1xk
ib1
rb1
22jk
x
2'2jk
x
2
02
22
'2
2
2
2
2
22
0
2)(;'2,2,1,)( knkkiknk xxixi where:
In each one of the three layers, we have zxy zxy EEEHH ,
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Plane Wave solutions in the three materials: TM case
x
z
y
zjxyy eeHzx
2
22 ),(H
Upper layer:evanescent plane wave
zjxjk
y
zjxjk
yy
eeH
eeHzx
x
x
1
1
1
11 ),(
H
Central layer:uniform plane waves
zjx
yy eeHzx 2'2'2 ),(H
Lower layer:evanescent plane wave
1xk
1xk
ib1
rb1
22jk
x
2'2 jkx
2
02
22
'2
2
2
2
2
22
0
2)(;'2,2,1,)( knkkiknk xxixi
where:
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Reflection Coefficients at the interfaces: TM case
x
z
y
22jk
x
2'2jk
x
( )
( )( )
G 12
12
21
1
1
/2
1
1... x
x
x
knnjtg
djk
y
djk
y
TM eeH
eH
( )( )
( )( )
( )
G
12
12
21
1
1
/2
21
2
12
21
2
12
21
2
12
21
2
12
12
12
1
1
/
/
/
/
coscos
coscos
x
x
x
knnjtg
x
x
xx
xx
ti
ti
djky
djk
y
TM
e
jknn
jknn
kknn
kknn
nn
nn
eH
eH
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x
z
y
The triangular points belongingto the same equi-phase plane must be
separated in phaseby an integer multiple of 2p.
The same for the square points.
Considering e. g. points A and C, itmust be:
A
'C
'
B''
B
''C
TM
CBjn
TMABjn
Ay
mj
AyCy
e
eH
eHH
G
G
'''01
'
01
''
1
2
11
p
Consistency Condition on Equiphase planes: TM case
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Dipartimento di ElettronicaInformatica e Sistemistica
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It must then be:
p
jm
knnjtg
djn
knnjtg
djn
e
e
e
e
e
x
i
x
ii
2
)/(2
cos
2
)/(2
2coscos
2
1
2
12
21
01
12
12
21
01
x
zy
A
'C
'
B''
B
''C
i
dCB
cos
2'''
i
i
dAB
2cos
cos
2'
+d
-d
)
(
'CB
span
''
)
('AB
span
TMG
TMG
Consistency Condition on Equiphase planes: TM case
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Characteristic Equation: TM case
22
)/(44
2)/(
44cos
24)2cos1(cos
2
11
2
1
22
1
2
12
21
1
1
2
12
2101
1
21
01
2
2
cos
222cos
cos
2
1
2101
1
2101
pp
p
p
p
mdktgdkn
ndm
knntgdk
mknn
tgdn
mk
tgd
n
eeeee
xx
x
x
x
i
x
i
i
jmk
jtgd
jn
k
jtgd
jn
xix
i
i
( )
( ) oddmdkdknnd
evenmdktgdknnd
xx
xx
,
,
11
2
122
11
2
122
cot)/(
)/(
We obtain then:
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u
w
p/2 p 3p/2
( ) ( )( ) ( ) )(cot)/(),(cot
)()/(),(2
12
2
12
oddoddeveneven
TMuunnwTEuuwTMutgunnwTEutguw
Resolution of TE and TM Characteristic Equations
From the characteristic equation we have:
Moreover, it must be:
TE0
TM0
TM1TE1TM2TE2
TM3
TE3
The graphic solution gives thevalues of
dwdkux 21
,
22 uvw
of the guided modes for a given valueof the normalized frequency:
2
2
2
1
2
2
2
10 )/( nndcnndv w
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Exploiting again the relations:
We can solve numericallythe equation to find (w)(e.g. TMcase withm=0):
and add the TMdispersion curves to the TE ones
2
02
2
2
22011
)()(
)()(
dndd
ddndkx
( ) ])()([)()(/)()(
22
01
22
01
2
12
2
02
2
ddntgddnnndnd
w
10
01
n
n
mw
20
02
n
n
mw
...
Dispersion curves (w): TE and TM case
(TE0)
(TE1)
(TM0)
(TM1)
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The following relations areexpoited:
The plot is more readableThe curves can be referred to waveguides different from each other
Only one TE and one TMmode propagate when it is 0
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Behavior of |Hy|for the modes TMm (m even, m=2k)
x
z
y
G
11
2222
)/(2
2
1
1
11
12
12
21
1
yy
djkkdkj
knnjtg
djk
y
yTM
HH
ee
e
eHH
xx
x
x
p
Considering for example the upper interface, we have from the characteristic equation:
In the central layer the field Hy is given by:
) ( ) zjxyzjxjkxjkyy exkHeeeHzx xx 1111 cos),( 11H
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Behavior of |Hy|for the modes TMm (m even)
x
z
d-d
|Hy1|Ex. mode TM0
x
z
d-d
|Hy1|Ex. mode TM2
( ) ( )xkHexkHzx xyzj
xyy 11111 coscos),(
HIn the central layerthe expression is given by
222
0 11ppp
xkdk xx
2
3
2
3
2
311
pppp xkdk xx
B h i f | | f h d TM ( dd 2k )
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Behavior of |Hy|for the modes TMm (m odd, m=2k+1)
x
z
y
G
11
22)12(2
)/(2
2
1
1
11
12
12
21
1
yy
jdjkkdkj
knnjtg
djk
y
yTM
HH
eee
e
eHH
xx
x
x
pp
Considering for example the upper interface, we have from the characteristic equation:
In the central layer the field Hy is given by:
) ( ) zjxyzjxjkxjkyy exkHeeeHzx xx 1111 sin),( 11H
B h i f | | f h d TM ( dd)
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Electromagnetic Technologies for Link Design M
Behavior of |Hy|for the modes TMm (m odd)
x
z
d
-d
|Hy1|Ex. mode TM1
x
z
d-d
|Hy1|Ex. mode TM3
( ) ( )xkHexkHzx xyzj
xyy 11111 sinsin),(
HIn the central layerthe expression is given by
pppp
xkdk xx 112
pppp 2222
3 11 xkdk xx
In general |Hy|of mode TMm assumesm times the `0` value for dxd
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Guided Modes and Radiation Modes22
01
2
1 )( nkx2
2
2
22
02
2
2 )( nkx
2
01 )( n
2
02 )( n
0
0
0
Planewavespropagatingalong z
Planewavespropagatingalong xin thecentral
layer
Plane
wavespropagatingalong xin theouterlayers
Planewavesattenuating
along z
Planewavesattenuating along xin the outer layers
GuidedModes
Radiation
modespropagatingalong z
Radiationmodesattenuating
along z
The set of guided and radiation modes forms a Complete Set of Solutionsof MaxwellsEquations for the dielectric slab Any field of the slab can be expressed as a linearcombination of these modes
Th O ti l Fib Cl ifi ti
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The Optical Fiber: Classification
SilicaFibres
PlasticFibres
Single Mode
Multimode
Multimode
Graded
Index
StepIndex
StepIndex
Graded
Index
StepIndex
Standard
DispersionShifted
P
ERFORM
ANCES+
COSTS
Long DistanceConnections
In-Building,Local AreaConnections
HomeConnections
S O
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Core: SiO2+ GeO210 mm (Single Mode),50, 62.5 mm (Multi Mode),n1~ 1.443
Cladding: SiO2125 mm
n2~ 1.44
Primary Coating
400 mm Secondary Coating1 mm
Silica Optical Fibres
Plastic Optical Fibers
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Core:Polymethylmethacrylate (PMMA),Perfluorinate (CYTOP), 62.5 mm 980 mmn1~ 1.49
Cladding): Other Polymericmaterial250 mm 1 mmn
2~ 1.41
External coating~2 mm
Plastic Optical Fibers
P d t fi d O ti l Fib G id d M d
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n1>n2
r
fn2
n1
a1. Resolution ofMaxwells Equations in the three regions
Solutions in terms of Bessel functions in the three
regions
2. Imposition of
continuity conditions of at the
interfacer = a
Homogeneous Linear System to be solved
Achievement of the Characteristic Equation (TE/TM,
even/odd)
3. Resolution of the Characteristic Equation
Modes propagation constant
Modes amplitude behavior
HEHE
,,, zz
)(w
Procedure to find Optical Fiber Guided Modes
S l i P d f M ll E ti
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zj
zzrr
zj
zj
zzrr
zj
eiHiHiHer,z)(r
eiEiEiEer,z)(r
ff
ff
)(),(,
)
(),(,
H
E
H
E
Evaluation of Ez, Hz:
0)(0)(
22
0
2
220
2
ziz
ziz
HnH
EnE
Evaluation ofEt, Ht:
)()(
1
)
()(
1
22
0
22
0
ztzizt
i
rrt
ztzzt
i
rrt
EijHjkn
iHiH
HijEjkniEiE
w
wm
H
E
r
fn2
n1
(2.1)
(2.2)
Solving Procedure of Maxwells Equations
(i=1 for core
i=2 for cladding)
Expressions of HE
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Where:
Jm ( ) = Bessel Function of First Kind of order m
Km ( ) = Modified Bessel Function of Second Kind of order m
Solving (2.1) (2.2) in cyilindrical coordinates yields:
r< a
r> a
r< a
r> a
2
02
22
2
22
01
2
1
n
nkt
Expressions of
r
fn2
n1
zzHE,
zj
m
zj
tmz
zj
m
zj
tm
z
emDmDr)(K
emCmCr)(kJ,z)(r
emBmBr)(K
emAmAr)(kJ,z)(r
f
f
)sin()cos()sin()cos(,
)sin()cos(
)sin()cos(,
212
211
212
211
H
E
E i f th Ch t i ti E ti
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Imposing the Continuity Condition the Characteristic Equation is obtained
Introducing the approximation D = (n12 - n22 ) / (2 n12 )
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Review on Bessel Functions of First Kind
)(0 xJ
)(1 xJ
Zeros of J0(x):2.40485.52018.6537
Zeros of J1(x):3.83177.015610.1735
Review on Modified Bessel Functions of Second Kind
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Review on Modified Bessel Functions of Second Kind
)(0 xK
)(1 xK
All Km(x)tend tofor x0 and fallexponentiallyfor increasing x.
The ratio Km(x)/ Km+1(x)
tend to 1 for increasing x
Both ratiosx*Km(x)/ Km+1(x)andx*Km+1(x)/ Km(x)are equal to 0
for x=0 andtend to x for increasing x
LP d ith 0 fi di k
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2 4 6 8 10
))((
))((
)( 21
20
2
1
2
12
1
2
akvK
akvK
akvt
t
t
)(
)(
10
111
akJ
akJak
t
tt
LP01 LP02
LP03
kt1a
LPmn modes with m=0: finding kt1
LP d ith 1 fi di k
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2 4 6 8 10
2 2
0 12 2
1 2 2
1 1
( ( ) )
( ) ( ( ) )
t
t
t
K v k a
v k a K v k a
0 11
1 1
( )
( )
tt
t
J k ak a
J k a
LP11LP12
LP13
kt1a
LPmn modes with m=1: finding kt1
O ti l Fib N li d Di i C
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2
2
2
10 nnav
2
2
2
0
2
1
2
0
2
2
2
0
2
nn
nb
Optical Fiber: Normalized Dispersion Curves
Behaviors similar
to the ones of the
dielectric slab
B h i f M d A lit d
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Linearly Polarized modes which are obtained utilizing the condition D
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Modes LP0n are two times degenerate: they can haveEt =Ex orEt =Ey
x
y
LP01LP02
Modes LPmn are 4 times degenerate:
They can haveEt =Ex orEt =Ey
They can have azimuthal dependence cos(mf) or sen(mf)
LP11
f
x
y
Degeneration of LP modes
Attenuation in Optical Fibers
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Attenuation in Optical Fibers
Source of
Information
Fiber Optic
Cable
Electrical
Signal
Destination
Optical
Source
Optical
Detector
Optical
Receiver
Attenuation in the Optical Fibers
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Intrinsic absorption
Extrinsic Absorption
Linear Diffusion
Attenuation in the Optical Fibers
Material: SiO2, SiO2+GeO2, SiO2+F,Atl~0.1 mm: electronic transitionsAt l~9. mm: photon interactionswith material molecular vibrations
Rayleigh Scattering: a part of the optical power is transferred from the propagationModes to other modes, including radiation modes, at the same frequency.It is due to slight inhomogeneities of the crystal over distances shorter than 1 mm
(Due to impurities)Interactions with bond vibrations ofOH-ion
Attenuation in SiO2 Fibers (SMF and MMF)
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Attenuation in SiO2 Fibers (SMF and MMF)
5
4
3
2
1
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
a(dB/km)
l (mm)
Rayleigh Scattering
OH- ionAbsorption
(This peak canbe reduced if
the fabricationprocedure isimproved)
Infraredabsorption
UltravioletAbsorption
Classical Transmission Windows
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Classical Transmission Windows
1 2 3
5
4
3
2
1
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
a(dB/km)
l (mm)
I windowl~ 0.8mm
II window
l~ 1.3mm III windowl~ 1.55mm
~ 0.2dB/km
~ 0.4dB/km
Recent Definition of new Transmission Windows
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Recent Definition of new Transmission Windows
I window
0.8mm
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Attenuation in Plastic Optical Fibers
500
400
300
200
100
0.45 0.5 0.55 0.6 0.65 0.7
a(dB/km)
l (mm)
250
200
150
100
50
0.6 0.8 1.0 1.41.2
a(dB/km)
l (mm)
Step-Index PMMA Graded-Index in Perfluorinated Polymer
Attenuation in Optical Fibres: Conclusions
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SiO2 Fibre
Attenuation in Optical Fibres: Conclusions
Utilizing appropriate optical transmission bands (III window, II window, etc),the attenuation is very low (a< 0.5 dB/km).
Long-distance links (unrepeateredspans of tens of km) Metropolitan networks
In-Building networks
The high attenuation (at present a> 30 dB/km) is one of the limiting factors forPOF links, which at present do not exceed tens or a very few hundreds of meters
of length.
Home networks Connections inside cars, trains, ships, etc
Plastic Fibre
Coupling Losses AC1 due to Numerical Aperture:1 N i l A t f th Fib
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1. Numerical Aperture of the Fibre
( )2
2
2
1sin nnNA MFFibre
Incindent rays which remain inside the AcceptanceCone of semi-aperture MF are guided inside thefibre
SMF : NAFibra~ 0.15MMF : NAFibra~ 0.30PMMF : NAFibra~ 0.50
Coupling Losses AC1 due to Numerical Aperture:2 N i l A t f th S
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( )MSSourceNA sin
2
101 log10Fibre
SourceC
SourceFibre
NANAA
NANAIf
For a fixed NASourcemultimodal fibres have lower values of AC1 withrespect to single mode ones
2. Numerical Aperture of the Source
Coupling Losses AC2due to Source Emitting Area:S E i l t Di t
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Source Fibre DCoreDSource
2
102 log10Fibre
SourceCSourceCore
D
DADDIf
SMF : DCore~ 10 mmMMF : Dcore~ 50-62.5 mmPMMF : Dcore~ 62.5-980 mm
For a fixed DSourcemultimodal fibersexhibit lower AC2values with respect tothe single mode ones
Source Equivalent Diameter
Definitions referred to a pulse signal
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x(t)
t
t
Rms puse width
(rms = root mean square)
Average pulse arrival time
( )
1/ 2
2
1 1( ) , ( )
( ) ( )
t tx t dt t t x t dt
x t dt x t dt
Definitions referred to a pulse signal
Dispersion in Optical Fibres
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Dispersion in Optical Fibres
Source of
Information
Fiber Optic
cable
Electrical
Signal
Destination
Optical
Source
Optical
Detector
Optical
Receiver
Types of Dispersion in Optical Fibres
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Types of Dispersion in Optical Fibres
Exists only in the MMF.
Each mode of the fiber carries with a different group velocity vgaportion of the modulating signal
Exists both in SMF and in MMF.
Each portion of the signal spectrum (wave packet) travels with adifferent vg.
Exists in SMF, can be neglected in MMF.
The signal is divided between the two degenerate x-polarized and
y-polarized LP01 modes. These modes ideally should have thesame vg, but in practice exhibit different vgs.
IntermodalDispersion
Chromatic
Dispersion
PolarizationModeDispersion
Dispersion in few words
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Dispersion in few words
The signal and/or its spectrum, exhibit an undesired subdivision in different portionsWhich travel with different group velocities vg. The complete denomination of thephenomenon is:
Group Velocity Dispersion (GVD)
Limitations at the Bit Rate of the link in case of digital transmission
Limitations at the Pass Band of the link in case of analog transmission
Consequences
Summarizing:
Intermodal Dispersion
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p
u= kt1a
Normalized frequency v high
u ~ 2.4048, 3.8317,5.1356,,That is, kt1,mn is independent from v from w
For modeLP01 it is kt1,01= 2.4048/(a)For modeLP11 it is kt1,11= 3.8317/(a)
For modeLP21 it is kt1,21= 5.1356/(a)
2 2
0 12 2
12 2
1 1
( ( ) )( )
( ( ) )
t
t
t
K v k av k a
K v k a
kt1,01a =2.4048
kt1,11a = 3.8317
kt1,21a = 5.1356
Computation of vg as dw/d
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( )
( ) ( )
mng
mnmnmnt
mnt
senn
c
d
dv
senc
nn
n
k
n
d
d
kn
w
m
mw
m
w
mw
1
1
2
01
2
10
2
2
,
2
10
2
,
2
10
2
cos11
n10
kt1,mn
n1
n2
n2
mn
For the fundamental mode it is vg = (c/n1) [1- (p/2a)2/(n1k0)
2]1/2 ~ c/n1
For the mode at cut-off it is sen mn =n2/n1vg~ (c n2/n12
)
For a span of lengthL the maximum difference between the arrival times is:
Dt=Ln1/c(n1/n2-1)~Ln1D/c
p g
Computation of vg through geometrical optics
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The same result can be obtained through geometrical optics:
DtAB= n1LiD/cDt= n1 {SLi}D/c=n1LD/c
Li
n1
n2
c
A
B
p g g g p
Pulse spreading and Bit rate
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Estimation of the transmittable bit rate:
Dt TB that is Br
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n(r)
(n2)
B
With the optimum value ofa (a ~ 2) it is:
Dt~(n1LD/c) (D/2)
B L=BrL~ (Gbit/sec) Km
a
n( r )=n1[1-2D(r/a)
a]1/2 if ra
A
Graded Index Multimode Fiber
Product Band x Distance
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Product Band x Distance
The effect of intermodal dispersion is represented by theProduct Band x Distance(~ Bit Rate x Distance)
Typical values:
Plastic Fibres
Step-Index Graded-Index
Silica Fibres ~800-1500 MHz Km~tens of MHz Km
~200-300 MHz Km~10-20 MHz Km
Chromatic Dispersion: main causes
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Waveguide Dispersion
The behavior ofis non linear
with wdue to the presence of the
two asymptotes=n2k0 and=n1k0
Material Dispersion
The fact that n1=n1(w) and n2=n2(w),
introduces a further cause of non lineardependence ofwith w.
w
1001 nkn mw
2002 nkn mw
(w)
...
1001 nkn mw
2002 nkn mw...
w...
p
Effects of Chromatic Dispersion
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Reduction of peak value andincrease of pulse duration
Limitation to the signal bit-rate
Digital Modulating Signal Analog Modulating Signal
Distortion of the signalLimitation of the pass band Numerical evaluation of distortion
Effects of Chromatic Dispersion
The harmonic components (WavePackets) which constitute the spectrum of the
Modulating signal arrive to destination with different delays
The Cromatic Dispersion is presnt also in the MMF but it can be oftenneglected with respect to intermodal dispersion
Chromatic Dispersion expressed in
kmnmpsec
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Electromagnetic Technologies for Link Design M
C o at c spe s o e p essed
D
kmnm
psecl
t
ll
d
d
d
vgd
Dg
1
Enlargement of a pulse for one km of fiber length for 1 THz of bandwidth
of the modulated signal.
tgt
g=v
g-1
(n
sec/m)
4.8784.875
DDl
Minimum of tgZero of DDl
kmnm
Chromatic Dispersion expressed in
kmTHzpsec
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D
kmTHz
psecw
t
ww
d
d
d
vgd
Dg
1
Enlargement of a pulse for one km of fiber length for 1 THz of bandwidth
of the modulated signal.
tgt
g=v
g-1
(n
sec/m)
4.878
4.875
DDw Minimum of tg
Zero of DDw
p p kmTHz
Material Dispersion in an homogeneous medium
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Electromagnetic Technologies for Link Design M
Normalized group delay:
Group velocity: vg=1/tg
Dispersion:
~
2
2
2
3
2
2
2
2
2 2...)2(2
1
)2(
1
lp
l
p
pw d
nd
cdf
nd
fdf
dn
cdf
d
D D
2
2
2
3
2
2
2
2
...)2(1
2
1
l
l
p d
nd
cdf
ndf
df
dn
cdf
dD f D
2
2
2
1
l
l
lpl
d
nd
cdf
d
d
dD D
p g
df
dg
pt
)2(
1~
Behavior of n(l)for pure silica
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Electromagnetic Technologies for Link Design M
32
0
5
22
0
42
0
33
02
2
0100
)()()(
)(
l
C
l
C
l
CCCCn
lll
lll
where:
C0=1.4508554
C1=-3.1268e-3 mm-2C2=-3.81e-5 mm
-3
C3=3.027e-3 mm2
C4=-7.79e-5 mm4
C5=1.8e-6 mm6
l=.035 mm2
(l0 in mm)
( ) p
Behaviors of and for pure silicalddn/22 / ldnd
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p
For l~1.3 mm it is (Zero Material Dispersion)
The behaviors are similar for doped silica
02
2
ld
nd
Ex. Of Material Dispersion: the RainBow
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p
b
180-2(90-2b+a)=4b-2aa
b
b
b
a
(180-4b)/2=
=90-2b
Sun at horizon horizontal raysRays all possible angles -90
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0 20 40 60 80 1000
10
20
30
40
50
Values of a around 60 degrees reflection angles very close to each other Intensity maximum.
50 55 60 65 737
38
39
40
41
42
43
Angle a (degrees)
Angle4b-2a(degrees)
The angle of the maximum varies with l: Red: 42.3 degrees, Violet: 40.4 degrees
Standard Fibres ITU-T G.652 e G.653
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G.652 : Standard Single Mode Fiber
Most used type of SMF Dispersion is zero for l~ 1.3mm
G.653 : Dispersion Shifted Fiber
More expensive, less used Dispersion is zero for l~ 1.55mm(III wndow C Band)
Chromatic Dispersion: Wave Packet
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( )
( )
D
DD
ft
ft
fE
ttx
p
p
w
sin
cos)0,(
0
0
D
D
D
Ld
dtf
Ld
dtf
fE
LtLtx
0
0
0
sin
cos),(
0
0
w
w
w
w
p
w
p
w
)0,(tx
t
),( Ltx
t
( ) LfjLfH exp),(
fD
f
0f0f
2/0
E
)0,()0,( fXfX
LfLfXArg )(),(
f
0f
0f
f
0f0f
2/0E
),( LfX( )F( )1-F
Chromatic Dispersion: Finite Bandwidth Signals
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( ) LfjLfH
exp
),(
)0,(tx
t
t0 ),( Ltx...
t0 +Dt
t
)0,()0,( fXfX
0f0f f
Lf
LfXArg)(
),(
),( LfX0f
0f
f
f
( )F( )
1-
F
Chromatic Dispersion: Gaussian Pulse for z=0
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( )tfeEtx Tt
02
0 2cos)0,(2
0
2
p
T0= rms pulse width
t
)0,(tx
0E
088.0 ET0
Chromatic Dispersion: Gaussian Pulse for z>0
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)2cos(
)2(
1),(
00
))2(
1(2
)2
1(
2
2
2
2
0
0
0
2
2
2
20
2
ztfe
zdf
d
jT
TAtzA
zdf
djT
zdf
dt
p
p
p
p
t
),( tzA
t
X(f,0)Multiplication by
)( fHfibre
( )tfeE
tx
T
t
0
2
0
2cos
)0,(
20
2
p
t
),0( tA
X(f,z)
( )F
( )1-F
Chromatic Dispersion: rms pulse width for z>0
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0
1
T
f DOptical source with low linewidth ( ):
2
2
2
2
0
2
0)
)2(
11()( z
df
d
TTzT
p
Optical source with linewidth not to be neglected ( ):
2
2
2
2
2
2
2
0
2
0)
2
1()
)2(
11()( z
df
dfz
df
d
TTzT
p
pD
0
1
Tf D
22
2
2
2
0
2
0)
2
1()
)2(
11()( z
df
d
d
dz
df
d
TTzT
lpl
pD
p p
Influence of dispersion on the Bit Rate
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p
Rule usually adopted :
5
)( BITt
zT
)(
2.0
zT
Br
Optical source with low linewidth ( ):
2
0
2
0)
1()( zDT
TzT wD
Optical source with linewidth not to be neglected ( ):
0
1
Tf D
0
1
Tf D
zDzdf
d
d
dzT ll
lpl
DDD
2
1~)(
zD
Brll DD
2.0
Minimum value for zDT wD0zD
Br
wD2
2.0
Chromatic Dispersion: Counter measures
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Use of Dispersion CompensatingFibers (DCF)
Use of Gratings(Ex. Fiber Bragg Gratings FBG)
Use of Soliton Propagation(exploiting the nonlinearities of the fiber)
(SMF +D) (DCF -D)
(SMF +D) (SMF +D)
(FBGD)
t t
Dispersion compensating Fibers
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Through a relatively high doping ofGeO2 it is possible to abtain anegative value ofD [ps/(nm km)]
L1(es G652)
L2
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Each channel needs a slightly different compensation
DG652
l(mm)
1.3 1.55
l(mm)
Spettro
WDM
In the band B it is:
DG652 B
~ D(l0) + (dD/dl) (ll
0)
It must then be:
DDCF B = -(L1/L2) DG652 B
=-(L1/L2) D(l0) -(L1/L2) (dD/dl) (ll0)
B
l0 l1DDCF
l(mm)
Pulse Enlargement per unit bandwidth
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It is represented the quantity
(ps/nm)
z
z
L1 L2 L1 L2 L1
Channel 1 WDM
Channel i WDM
Channel N WDM
Ideal Case
Real Case
z
dD0
)(
z
dD0
)(
z
dD0
)(