CHAPTER 4
Fiber optic III: attenuation and dispersion
SM Olaizola
2004
Outline
Attenuation
Absorption
Scattering
Radiative losses
Dispersion
Material dispersion
Waveguide dispersion
Other sources of dispersion
Optical
link calculation
Attenuation
Definition: a loss of signal strength in a lightwave, electrical or
radio signal usually related to the distance the signal must
travel.
Determines the maximum transmission distance between
transmitter and receiver.
Attenuation is caused by:
Absorption
Scattering
Radiative losses (bending losses)
Intensity
Intensity
n2
t
n1
t
Attenuation
It is normally assumed that a decrease in the
signal is proportional to the length of the fibre and
the signal power (Beers Law)
dP = −α p Pdx
Integrating
1  P(0 ) 

P (z ) = P(0 )e
→ α p = ln
z  P (Z ) 
This is the attenuation coefficient given in km-1. It
can be changed to dB/km with
10  P (0 ) 
 = 4.343α p km −1
α (dB/km ) = log
z
 P(z ) 
−α p z
(
)
Attenuation:absorption
Absorption
is caused by three different
mechanisms:
Absorption by atomic defects.
Extrinsic absorption by impurities
Intrinsic absorption by constituent atoms
Attenuation: absorption
Impurity absorption:
Impurities are in the form of
transition metals: Fe, Cr, Cu,
Co and –OH ions:
Transition metals show
broad absorption peaks.
OH ions, produced during
the fabrication of the fibre
show large absorption peaks
at 1400, 950 and 725nm.
Good quality fibres require
that only ppb of impurities are
present in the fibre
From Keiser’s Optical
fiber communications
Attenuation: absorption
Intrinsic absorption is
associated with the basic
fibre material (SiO2).
Intrinsic absorption sets
the fundamental lower
limit on absorption:
In the UV region absorption
occurs when a photon
excites an electron of the
valence band to a higher
energy level.
In the IR region, absorption
occurs due to resonant
vibration of atomic bonds in
the fibre.
From Keiser’s Optical
fiber communications
Attenuation: absorption
In numbers:
UV absorption follows Urbach’s law
αUV = C E E
E,C are empirical constants
For Ge doped SiO2 fibres, the UV absorption
depends on the Ge mole fraction x
αUV
0
154.2 x
 4.63 
−2
=
10 exp

46.6 x + 60
 λ 
In dB/km
A empirical expression that describes IR absorption
for GeO2 - SiO2 fibres is
 − 48.48  In dB/km
11
α IR = 7.81·10 exp

 λ 
Attenuation
Question: why the sky is blue?
Rayleigh scattering!!!
Attenuation: scattering
Scattering
losses in fibres arise from:
microscopic variations of material density
and
microscopic variation of dopant density.
These
effects produce a variation of the
material refractive index in a smaller
scale than the wavelength which causes
Rayleigh-type Scattering.
Attenuation: scattering
The equations that describe the
scattering are:
For scattering arising from
density fluctuations
α scat
2
8π 3 2
= 4 n − 1 k BT f β T
3λ
(
)
For multicomponent glasses
the scattering is:
α scat =
2
8π 3
∂n 2 ∂V
4
3λ
(∂n )
m
 ∂n 2 
 ∂n 
2
(∂Ci )2
=   (∂ρ ) + ∑ 
i =1  ∂Ci 
 ∂ρ 
2 2
( )
2
kB is boltzmann constant
Tf temperature at which
density fluctuations are
frozen
βT is the compressibility
of the material.
∂ρ is the density
fluctuation.
∂ Ci is the fluctuation of
the concentration of the
ith element,
Attenuation
As a rule of thumb:
•At 1310nm: 0.5dB/km
•At 1550nm: 0.3dB/km
From Gowar’s Optical
communication systems
Attenuation: radiative losses
Radiative
losses: also called bending
losses, occur when the fibre is curved.
There are two types of radiative losses:
Microbending losses.
Macrobending losses.
Absorption: radiative losses
Macrobending
Field
distribution
Microbending
(cabling losses)
a
Produces coupling to higher
order modes (a) or power loss (b)
R
R
(a)
Modes supported by fibre
decreases:
N eff
 α + 2  2a  3  2 3  
  
 + 
= N ∞ 1 −
2
α
∆
R
2
n
kR

 2   

(b)
Dispersion
What happens to the
shape of an optical pulse
when it travels along the
fibre?
From Saleh’s Photonics
Dispersion
The phenomenon in an optical fibre whereby light
photons arrive at a distant point in different phase
than they entered the fibre.
Dispersion causes receive signal distortion that
ultimately limits the bandwidth and usable length of
the fibre cable.
The two main causes of dispersion are:
Material (Chromatic) dispersion
Waveguide dispersion
Intermodal delay (in multimode fibres)
Intensity
Intensity
n2
σi
t
n1
σo
t
Dispersion
Note that dispersion
as defined in optical
fibre is different as
the dispersion
defined in optics
Doptics =
D fibre
dn
dλ
1 dTG d 2 n
=
∝ 2
L dλ
dλ
Optical dispersion as envisioned by
Pink Floyd
Dispersion
From Keiser’s Optical fiber communications.
Dispersion
Normally a pulse
that propagates
through an optic
fibre can be
assumed to be
Gaussian.
Pm a x
Pmax
2σ
e
Pmax
2
∆t1/2
t2
A
2σ 2
f (t ) =
e
σ 2π
For a Gaussian pulse σ=0.425∆τ1/2
t0
t
Dispersion
The parameter s is used to characterize
the pulse-width. If we use this parameter
then the intramodal (material) and
intermodal dispersion can be easily added
σ
2
TOTAL
=σ
2
inter
+σ
2
intra
Watch out: The pulse width can be defined
both in the wavelength domain σλ or in the
temporal domain σt.
Dispersion: material
Material dispersion is caused by
the wavelength dependence of
the refractive index.
Each spectral component travels
independently and undergoes a
different group delay .
Formally, the diffusion parameter
D is obtained by deriving the
group delay with respect to λ.
D(ps/nm·km) =
1 dTG
= DM + DW
L dλ
Dispersion: Material
The material dispersion is caused by the
dependence of the group velocity with wavelength
(ultimately caused by n(ω)).
VG =
The time delay over a distance L is
TG =
c0
c0
=
N G n − λ dn
dλ
L LN G
=
VG
c0
And finally the material dispersion DM is
1 dTM
1 dN G 1 d 
dn 
λ d 2n
=
=
DM =
n −λ
=−
L dλ
c0 dλ
c0 dλ 
dλ 
c0 dλ2
Dispersion: Material
This graphic shows the DM calculated with the
Sellmeier equation for SiO2
SiO2
Dm(ps/nm·km)
Wavelength (µm)
Dispersion: Material
But how is DM related to the time spreading?
Approximating the differential equation with
increments
σ t ≈ DM Lσ λ
If DM <0 then shorter wavelengths need more time to
reach their destination (normal dispersion).
If DM >0 then longer wavelengths need more time to
reach their destination (anomalous dispersion).
Dispersion: material
Note: the dispersion analysis can also be
performed with the definition of group velocity
in terms of β:
2
L  dβ
2 d β 
 2λ
δλ
δτ =
δλ = −
+λ
2 
dλ
2πc  dλ
dλ 
dTg
Or in the frequency domain
d  L
δτ =
δω =
dω
dω  Vg
dTg
 d 2β 
 2  = β 2
 dω 

 d 2β 
δω = L
 dω 2 δω = Lβ 2δω




is the group velocity
dispersion (GVD)
parameter
Dispersion: waveguide
Waveguide
dispersion arises
from the b
dependence on
V of the
waveguide:
V=V(λ)
b=b(β)
From Buck’s Fundamentals of optical fibers
Dispersion: waveguide
Let’s try to quantify this dispersion. From
the definitions of b and V (see chapter 3):
b=
β 2 k 2 − n22
2
1
2
2
n −n
1 ≈ n2
n
→ b =
β k − n2
n1 − n2
(
)
2 12
2
2
1
V = ka n − n
→ β ≈ kn2 (1 + ∆b )
Therefore, the group delay caused by
waveguide dispersion can be expressed
as:
L
L dβ L 
d (kb ) 
T =
=
=
n +n ∆
W
VG
c0 dk

c0 
2
2

dk 
Dispersion: Waveguide
Also note that
d (kb ) d (Vb )
=
dV
Therefore dk
L
L dβ L 
d (Vb ) 
TW =
=
=  n2 + n2 ∆

VG c0 dk c0 
dV 
And finally as the dispersion is
DW =
1 dTW dTW dV
=
L dλ
dV dλ
Dispersion: waveguide
Calculating both derivates:
 2πa

d
NA 
dV
λ
 = − 2πaNA = − V
= 
dλ
dλ
λ2
λ
dn/dλ=0
L
d (Vb ) 
d   n2 + n2 ∆

c
dV
dTW
 n2 L d 2 (Vb )
0 

=
≈
∆
dV
dV
c0
dV 2
Therefore, finally:
V n2 L d 2 (Vb )
n2 L  d 2 (Vb ) 

DW = −
∆
=−
∆V
2
2 
λ c0
dV
λc0  dV 
Dispersion: waveguide
The figure shows the
behaviour of the first
and second derivates.
An approximate
function for the LP01
can be calculated by
using the expression
(1 + 2 )
b(V ) = 1 −
2
1

4 4
1 + 4 + V 


(
)
2
From Keiser’s Optical communication systems
Dispersion: Wavelength
The total chromatic dispersion of the fibre is
shown in the figure:
Dispersion: other sources
Profile dispersion: caused by the
dependence of the group velocity on ∆.
Typically less than 1ps nm-1km-1.
Therefore, the total chromatic dispersion is
D = DM + DW + DP
Dispersion: other sources
Polarization
mode dispersion (PMD):
caused by non-symmetry or
inhomogeneities of the fibre.
The result is that the propagation
constant for different polarizations is also
different.
Typically the dispersion caused by PMD
is less than a fraction of a picosecond
per kilometre. (but scales with L2).
Dispersion: dispersion shifted fibres
It is be desirable to minimize dispersion at the
wavelength of propagation. The Zero
dispersion fibre is conveniently placed near
1310nm for SiO2 fibres. But what if we are
using the third window?
Also for multiple channel communication it
may be desirable to have near zero dispersion
for a range of wavelengths
What is the solution?
Doping: increases attenuation
Change the waveguide geometry
Dispersion: dispersion shifted fibres
Changing the
waveguide geometry,
changes the shape of
the fundamental mode,
and so changes the V/b
and DW.
The chromatic
dispersion is normally
reduced between 1-3
ps/nm·km in a range of
wavelengths from 1.3 to
1.6µm
This may also lead to a
small increase in the
attenuation.
From Kasap’s Optoelectronics and photonics
Dispersion: intermodal dispersion
Caused by different modes having different propagation
constants in multimode fibres.
When the number of modes is high, it can be approximated
estimating the different path lengths from the fastest and the
slowest propagating mode.
Total length travelled by the fastest propagating mode: L0
Total length travelled by the fastest propagating mode:
L0/sinθc.
Therefore
L sin θ C
L
n∆
∆t ≈ 0
− 0 ≈ 1 L0
c0 n1
c0 n1 c0
Dispersion: intermodal dispersion
For GRIN fibres, the delay between the lowest
order and highest order propagation mode can
be approximated by:
σ inter
 n1∆L  α − α opt 

 for α ≠ α opt

c
+
2
α


=
2
∆
L
n

1
for α = α opt

c
With α being the parameter that defines the
index profile and αopt≈ 2
Optical link
When designing an optical link one of the most
interesting questions is the maximum distance
between emitter and detector for a certain bit
rate.
Two different analysis have to be done:
An analysis of the attenuation budget: Which is the
maximum distance before the signal is too small
and the photodiode cannot detect it? (attenuation
limited link)
An analysis of the dispersion budget: which is the
maximum distance before the optical pulse
broadens beyond the value when they overlap?
(dispersion limited link)
Optical link: attenuation budget
The attenuation budget of the fibre can be calculated
by substracting the attenuation of all the
components in the fibre and the photodetector losses
as follows:
(α
F max
= α emitter − α CL − ∑ α IL − α Phdiode )dB
Lmax =
α F max
α
Where
αemitter id the gain of the emitting device.
αCL is the coupling losses.
αIL are the insertion losses of each component in the fibre
αPhdiode is the sensitivity of the photodiode
Optical link: dispersion budget
The bit rate capacity is directly related to the
dispersion characteristics as the pulse
spreading is the limiting factor for a
maximum rate over a distance L.
From Kasap’s Optoelectronics and photonics
Optical link: dispersion budget
Note that the optical bandwidth is defined different
than the ‘traditional’ electrical bandwidth
(P0 /Pi ) as a function of modulation frequency
Electrical signal (photocurrent)
Optical Bandwidth for
Gaussian Dispersion
0.707
σ : total rms dispersion through the fiber
Fiber
Sinusoidal signal
1 kHz
Emitter
t
fel
1
1 MHz
1 GHz
f
Photodetector
Optical Output
Optical Input
Sinusoidal electrical signal
f = Modulation frequency
Pi = Input light power
Po = Output light power
Po /Pi
1
fop
50% below
0.5
0
t
0
t
1 kHz
1 MHz
1 GHz
f
Optical link: dispersion budget
Also the bit rate can be:
Non-return to zero (NRZ)
when the light signal does not
have to return to zero between
consecutive pulses..
Return to zero (RZ) bit rate
when the light intensity has to
return to zero before a second
pulse arrives Therefore:
BNRZ=2BRZ
It can be shown that light
pulses had to be separated by
at least 4σ to distinguish
between pulses
I
NRZ
t
I
RZ
4σ or 5σ
0.61
t
2σ
Optical link: dispersion budget
The maximum bit rate is
(BRZ )MAX
≈
1
4σ t
But the time spreading of the pulse depends
on the dispersion parameter of the fibre
σ t ≈ D Lσ λ
Therefore (B ) L ≈ 0.25
RZ MAX
D σλ
Optical link: dispersion budget
From the same expression, the maximum
dispersion-limited length in a fibre can be
written as
LMAX
0.25
≈
D σλB