(2)

Solutions W086/085/904

1 / (a) / If q must be greater than qc, the critical angle, for TIR and thus propagation to take place, then the maximum value of q1 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.
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:
(Assumes refractive index of air is approx 1.)
[16 Marks]
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 dt 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 angle can be calculated by the following formula.
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 the average of the four numerical aperture values to yield a final result. Now repeat the above procedure for the remaining fibres.
(i)  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.
(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]
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) Triangular profile
(iv)  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 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]
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:
Where d is the lateral misalignment. Worst case occurs where a, the radius is smallest (49.5 µm diameter core).
For a loss of 0.077 dB, the value of d is 0.692μm.
[14 Marks]
2 / (d) / (i)  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
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:
Pin = 60 µW and Pout = P3 + P4 = 54.5 µW so the excess loss is 0.42 dB.
Insertion losses:
Port 1 to 3: 4.40 dB
Port 1 to 4: 2.63 dB
Crosstalk:
Using P1 = 60 µW and P2 = 0.005 µW the results is -40.8 dB.
Split ratio:
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 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:
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]
3 / (e) / Effect of Moisture