Solutions W086/085/904
(2)
1 (a)
Cladding


Fibre Axis

Core
If  must be greater than c, the critical angle, for TIR and thus propagation to take
place, then the maximum value of 1 under these circumstances is the fibre acceptance
angle. Visualised in space the acceptance angle is defined is a conical half angle, for
the fibre. The numerical aperture for a fibre is the sine of the acceptance angle.
The analysis below applies only to rays entering along the axis, so called meridional
rays.
Assuming  is equal to c, the critical angle, what is the value of 
But
and by Snells Law
so
now
 2  90   c
n0 sin  1  n1 sin  2
 1  sin 1(n1 sin(90  c ))
 1  sin 1(n1 sin(90  c ))
 1  sin 1 n 1
cos 2  c

 1  sin 1 n 1

hence
n0 is the refractive
index of air =1
n 22
1 n 2
1
Simple trigonometry



Simple trigonometry
 1  sin 1  n 12  n 22 
This last value is the maximum value that can take on if TIR is to take
place, it is therefore called the fibre acceptance angle.
Numerical aperture (NA) is defined as the sine of the acceptance angle for a fibre.
From the analysis above the NA can be written as:
NA  (n12 - n22 )
(Assumes refractive index of air is approx 1.)
[16 Marks]
(3)
Solutions W086/085/904
1 (b) Based on results in part 1(a) we can state that: As the acceptance angle for a fibre
increases the numerical aperture also increases.
And based on the formula for worst case pulse spreading: As the numerical aperture
increases the time delay spread t for a pulse input also increases, in effect the modal
dispersion increases. It is known that an increase in dispersion reduces the bandwidth
of a fibre.
Overall then it is apparent that there is significant trade-off between a high acceptance
angle and a large fibre bandwidth. The impact of this trade-off in an actual optical fibre
system is important, for good coupling efficiency for optical sources such as lasers or
particularly LEDs a high fibre acceptance angle is very useful. But in effect this means
that by selecting a fibre with a high NA value (eg. plastic optical fibre) and thus
improving the coupling efficiency the result will be a reduction the fibre bandwidth. In
simple terms the trade-off could be defined as one between transmitted (launched)
power and bandwidth. As the transmitted power level influences the received power
levels this means that in pure communications terms a difficult trade-off between
received power and transmission bandwidth can arise for system designers.
[10 Marks]
1 (c) EQUIPMENT:






HeNe. Laser, emitting at a wavelength of 632.8 nm
50/125 graded index silica fibre.
200/400 step index polymer clad silica fibre.
1 mm core step index plastic fibre
Circle target, 5 mm spacing
Vernier Callipers (Ruler)
PROCEDURE:
This exercise is best carried out in semi-darkness. The numerical aperture is a
parameter normally associated with light entering a fibre, however to measure
numerical aperture it is easier to investigate the characteristics of the light leaving the
fibre, which will provide a reasonable approximation of the numerical aperture.
Important: Do not look into the light beam emitted from the laser or the fibre at
any time, as this could cause eye damage.
Connect one of the fibres to the HeNe laser, noting that it is easier to start with one of
the fibres with a large core diameter, such as a 1mm core plastic fibre. Take the other
end of the fibre and project the light output on to the 5mm circle target provided.
Determine the circle diameter R of the light and the distance L from the fibre end to the
circle as in the diagram below. Hence calculate the acceptance angle and by taking the
sine of the acceptance angle find the numerical aperture of the fibre. The acceptance
R
angle can be calculated by the following formula.   Tan 1
L
Now for the same fibre repeat this procedure for at least four other values of distance
L, calculating the acceptance angle and numerical aperture in each case. Finally take
Solutions W086/085/904
(4)
the average of the four numerical aperture values to yield a final result. Now repeat the
above procedure for the remaining fibres.
Circular Target
R
Optical fibre
(i)
θ
L
The fibre core diameter does not influence the fibre NA. The NA is only
influenced by the refractive index of the core and the cladding as described
in the following equation.
2
2
NA   n1  n2 


(ii)
Measuring L accurately is difficult.
The light radiated on the screen from the fibre is diffracted and hence there is no clear
edge making the measurement of R subjective.
The optical fibre is not a point source making the value R larger.
Assumes the optical power is evenly distributed across the core.
[14 Marks]
1 (d) (i)
A tight jacket cable would usually be used in a building installation. Tight jacket cables
are capable of tighter bend radii than loose tube cables. Also loose tube cables usually
have a water blocking gel in the tube that will pour out in a vertical installation.
(ii)
Tensile strength – A strength member to stop the cable being stretched while installed
Crush resistance – To stop the weight of the soil damaging the fibre.
Moisture ingress – To stop water coming in contact with the fibre and causing it to fail.
[10 Marks]
(5)
Solutions W086/085/904
2 (a) The refractive index profile of a fibre is define as the relationship between refractive
index and distance, in a radial fashion, from the centre of the core outwards to the
cladding and beyond. There are a number of common refractive index profiles:
(i) Step index
Step index is the simplest profile, commonly found in large core multimode fibre,
suited to low bandwidth, short range applications
(ii) Graded index
In a graded index core the refractive index follows a parabolic like form out from the
centre of the core to the cladding. The result is lower modal dispersion. It is used in
multimode fibre types for modest campus types distances were the limited bandwidth
(typically 200-800 MHz.km) is acceptable
(iii) Depressed cladding
Depressed cladding profiles are a feature of some singlemode fibres, as opposed to a
conventional matched cladding profile. The advantage of this profile is a reduced
susceptibility to bend loss. Typically such fibre are used over longer distances, but
where bend loss is an issue.
(iii)
(iv)
(iv) Triangular profile
A triangular profile is found in some singlemode fibres which are dispersion shifted.
This means that the dispersion minimum no longer occurs in the 1320 nm window, but
instead in the 1550 nm window. Typically used over long distances at high bit rates.
[10 Marks]
2 (b) In a fibre-to-fibre joint fibre core misalignment represents loss due to the mechanical
limitations and tolerances of the jointing technology. Typical examples are:
End separation loss
Loss caused by a gap between the ends of the two fibres in a joint. Power loss results
from Fresnel reflection and beam spreading.
Lateral misalignment
This one of the most problematic forms of loss. As shown in the diagram below it
(6)
Solutions W086/085/904
results from misalignment of axes of the fibre cores.
Even if a fibre to fibre joint is mechanically perfect slight differences between the
fibres, so-called fibre parameter mismatch can give rise to loss or attenuation at the
joint. Typical examples are (two only required):
•
•
•
•
Differences in numerical apertures
Core diameter differences
Core concentricity
Core ellipticity
Demountable connectors involving an air-gap:
In such connectors the presence of an air-gap would indicate a low quality connector.
For this reason loss is likely to be dominated by fibre core misalignment mechanisms
such as:
•
Fresnel loss
•
Beam spreading
•
Lateral misalignment
A fusion splice using 3-axis fibre positioning
In this type of joint fusion splicing will eliminate any air-gap and since 3-axis
positioning is used lateral and angular misalignment are unlikely. The three most likely
causes of loss are therefore fibre parameter mismatches such as:
•
•
•
Differences in numerical apertures
Core diameter differences
Core ellipticity
Core concentricity has been improved significantly on production fibres over the last
few years to allow the use of singlemode single axis fusion splicers so for this reason it
is not cited.
A mechanical splice.
By comparison with the previous case in this type of joint the fibre cladding rather than
the core is aligned. For this reason loss is likely to be a mix of fibre core misalignment
and fibre parameter mismatch mechanisms. Because of the presence of an index
matching gel in the joint any air gap will not give rise to Fresnel or beam spreading
losses so these are not included. The most likely causes of loss are:
•
•
•
•
Lateral misalignment
Differences in numerical apertures
Core diameter differences
Core ellipticity
[12 Marks]
Solutions W086/085/904
(7)
2 (c) Core diameter d is 50 µm  0.5 µm. For no other loss sources the worst case loss will
occur when the fibre diameters are at opposite end of the diameter tolerance band, that
is 50.5 µm and 49.5 µm, assuming that the output fibre has the smaller diameter.
The approximate loss is given by:
-20 log (D1/D2) = -20 log (50.5/49.5) = 0.173 dB
If lateral misalignment is also present then loss will increase. If the total worst case loss
is to be less than 0.25 dB, then the loss due to lateral misalignment must be <0.077dB
0.25 dB - 0.173 dB = 0.077 dB
The loss due to lateral misalignment is given the approximate formula:
  2d  
 10 Log 1   
  a 
Where d is the lateral misalignment. Worst case occurs where a, the radius is smallest
(49.5 µm diameter core).
 
2d

0.077dB  10 Log 1  
6 
   24.75 *10 
 
2d

0.0077dB  Log 1  
6 
   24.75 *10 
 
2d

10 0.0077  1  
6 
   24.75 *10 
 
2d

1.0178  1  
6 
   24.75 *10 
2d


1.0178  1  
6 
  24.75 *10 
0.0178 *  24.75 *10 6  2d


0.0178 *  24.75 *10 
d
6
2
d  0.692m
For a loss of 0.077 dB, the value of d is 0.692μm.
[14 Marks]
(8)
2 (d) (i)
Solutions W086/085/904
Essay style answer covering the following points
System a has a lower bandwidth than system b because
- it has a larger core size, hence greater modal dispersion
- its NA is higher which also increases its modal dispersion
- system b operates at 1330 nm which is minimum for material dispersion
Also a laser will have lower material dispersion than an LED due to its narrower
spectral width.
(ii)
Using a 850 nm source over single mode fibre is unlikely because
- Of the high attenuation
Using a 850nm or 1330 nm LED or over single mode fibre is unlikely because
- Of the difficulty in coupling light into the fibre because of the small core size
and low NA
[14 Marks]
3 (a)
Four port coupler used as a splitter for illustration, with power applied to port 1. Power
in and out of ports is given by P1, P2 etc..
Split Ratio (or Coupling Ratio)
Defined as the percentage division of optical power between the output ports. Most
common split ratio available is 50%,
i.e. P3 = P4. Split ratio frequently given as X/Y, where X and Y are the output ports.
Insertion Loss (dB)
Defined generally as the loss between two particular ports. It depends on the splitting
ratio and any imperfections in the coupler (excess loss). For example for a 50/50 split
ratio a reasonable insertion loss is < 3.4 dB
Excess Loss (dB)
In an ideal coupler the insertion loss would be set by the splitting ratio only,
so for a 50/50 splitter, the insertion loss would be 3 dB. In practice scattering,
absorption and imperfections raise the loss above the theoretical value. This is the
so-called excess loss. The excess loss for a particular port is the difference
between the ideal insertion loss calculated from the split ratio and the measured
insertion loss. For example for a 50/50 split ratio a reasonable excess loss is < 0.4 dB
(9)
Solutions W086/085/904
Crosstalk or Directivity (dB)
Crosstalk is a measure of the isolation between two input or two output ports. Power
from port 1 may be backscattered to port 2 and vice-versa. Crosstalk is the dB log ratio
of the power into port 1, by comparison with the power leaving pot 2. For a quality
50/50 splitter a typical crosstalk is normally less than -60 dB. Directivity is
crosstalk written as a positive value.
[8 Marks]
3 (b) Let the port powers be P1, P2, P3 and P4.
Excess loss:
P
i
n
E
x
c
e
s
s
L
o
s
s
=
1
0
l
o
g
d
B
P
o
u
t
Pin = 60 µW and Pout = P3 + P4 = 54.5 µW so the excess loss is 0.42 dB.
Insertion losses:
Port 1 to 3:
Port 1 to 4:
4.40 dB
2.63 dB
Crosstalk:
P
2
C
r
o
s
s
t
a
l
k
f
r
o
m
C
r
o
s
s
t
a
l
k
=
1
0
l
o
g
d
B
P
o
r
t
1
t
o
P
o
r
t
2
P
1
Using P1 = 60 µW and P2 = 0.005 µW the results is -40.8 dB.
Split ratio:
P
3
S
p
l
i
t
R
a
t
i
o
=
x
1
0
0
%
P
3
+
P
4
The value of the split ratio is 40% / 60%
[12 Marks]
3 (c) Fibre is intrinsically very reliable in a benign environment with few documented failure
mechanisms. Most failures are caused by poor cable choice, poor installation or
accidental damage. The intrinsic tensile strength of fibre exceeds that of an equivalent
steel wire so that the theoretical strength is 20 GPa (2,900,000 Psi). However due to
surface defects such as cracks strength in practice is much lower, typically 5 GPa (725
kPsi).
All produced fibres are proof tested after production using the apparatus sketched
(10)
Solutions W086/085/904
below. Between the constant speed motor and the constant torque motor the fibre is
stressed to typically three times the normal service maximum. A weak fibre with large
surface defects present after production will break under the applied stress and is not
subsequently used within cables. A typical industry test involves a stress of 50 kPsi
(350,000 kPa).This test guarantees a maximum surface defect size of 2.3 microns. The
predicted lifetime at maximum service stress with such a defect size present is 30
years.
Sketch of Proof Test Apparatus
[10 Marks]
3 (d) The time-to-failure tf in a fibre is normally assumed to be given by:
n
t
=
A
s
f
where A is a constant and n is the so called fatigue susceptibility parameter. To
estimate the lifetime at the maximum service stress of 0.1 GPa it is necessary find the
constant A for the fibre. Given the proof test data and assuming that the proof test
stress would have resulted in fibre failure just after 1 second then the time-to-failure
can be set to 1 second.
Thus rearranging the equation above in log form:
log 1 = log A - n.log(0.45 GPa)
Substituting for n=18.1 gives log A = 156.62
If the maximum stress is now assumed then the log of the time-to-failure is:
log tf = 156.62- n.log(0.14 GPa)
log tf = 156.62-147.44 = 9.17
The time-to-failure is 1.49 x 109 seconds or about 47.45 years.
[10 Marks]
(11)
Solutions W086/085/904
3 (e) Effect of Moisture
Moisture does not penetrate silica glass, so it does not affect propagation. But the
presence of water as OH ions on the fibre surface accelerates crack growth. This
process is called stress corrosion. Moisture protection is therefore of critical importance
in fibre cables.
Effect of Temperature
At 90 degrees centigrade the fatigue susceptibility parameter is less than half that at 25
degrees, because higher temperatures increase the rate of the stress corrosion chemical
reaction. The result is that fibre strength decreases by 25% at 90 degrees compared to
25 degrees. High tensile strength and zero moisture ingress cables are therefore
essential at elevated temperatures
[10 Marks]