Questions
1 (a)
Explain with the aid of a diagram what is meant by the terms acceptance angle and
numerical aperture for an optical fibre. Hence derive an expression for the numerical
aperture of a step index fibre with a core refractive index of n1 and a cladding
refractive index of n2.
[6 Marks]
1 (b) For a singlemode optical fibre explain concisely what meant by the terms mode field
diameter and normalised spot size.
[6 Marks]
1 (c) For a singlemode fibre the mode field radius w is related to the fibre core radius a by
the expression:
- 3/2
-6
w
= 0.65 + 1.619V
+ 2.879V
a
For a given optical fibre at a wavelength of 1550 nm the mode field diameter is 9.8
µm, when the normalised frequency is 2.1. Using this information find the core radius
of the fibre and the fibre numerical aperture. Hence determine if singlemode operation
is still possible at 1320 nm for this fibre. Comment briefly on the decision with
reference to the two possible definitions of cut-off wavelength in common use.
[13 Marks]
Questions
2 (a) Explain clearly what is meant by material dispersion in an optical fibre What are the
factors which influence the level of material dispersion in a particular transmission
system and how can material dispersion be reduced?
[6 Marks]
2 (b) An optical source has an r.m.s. spectral width of 3.1 nm, at a centre wavelength of
1550 nm.
The dimensionless material dispersion coefficient Y of the fibre is defined by:
d 2 n1
Y
d2
2
And the material dispersion coefficient is defined by:
 d 2 n1
Dc 
c d 2
where n1 is the core refractive index. If Y has a value of 0.01 show that the fibre
material dispersion parameter is 22 ps/nm.km. Hence determine the fibre span in km at
which the pulse broadening will exceed 3 ns.
[8 Marks]
2 (c) Describe an experiment to measure the light current characteristics of a Vertical Cavity
Surface Emitting Laser diode (VCSEL) stating any precautions that may be taken.
Show how the following characteristics can be determined.
1. The lasing threshold
2. The differential quantum efficiency (above and below threshold)
[10 Marks]
Solutions
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
co s 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.)
[6 Marks]
Solutions
1 (b) Mode field diameter (MFD) is an important property of SM fibres. The amplitude
distribution of the HE11 mode in the transverse plane in the fibre is not uniform, but is
approximately gaussian in shape. The MFD is defined as the width of this amplitude
distribution at a level 1/e (37%) from the peak or for power 13.5% from the peak
The spot size is the mode field radius w. Its value relative to core radius “a” is given by
the expression:
3
/
2
6
w

=
0
.
6
5
+
1
.
6
1
9
V
+
2
.
8
7
9
V
a
where V is the fibre “V-value”.
[6 Marks]
1 (c) Firstly it is noted that the two unknown parameters are the fibre core radius and the
numerical aperture.
Firstly calculate the actual fibre core radius “a”. Using the formula relating w/a to the
normalised frequency V then since V is 2.1 then w/a is 1.22.
The MFD is 9.8 µm, so the spot size is 4.9 µm. Thus the core radius is 4 µm. Using
this value of core radius and the normalised frequency V of 2.1 it is possible to also
find the numerical aperture for the fibre:
2

V
=
a
.
N
A

For a wavelength of 1550 nm from this expression the NA is found to be 0.128
To determine if singlemode operation is possible at 1320 nm we must find the fibre
cutoff wavelength. Using the all of the parameter values available the fibre cutoff
wavelength can be calculated by rearranging the expression above and using the cutoff
value of the normalised frequency V (= 2.405).
2

a
N
A
c
=

V
c
Using the above expression the cutoff wavelength is found to be 1353 nm. Below this
wavelength V > 2.405, so singlemode operation is not possible at 1320 nm according
to the strict criteria above.
In practice the theoretical cutoff wavelength above is difficult to measure. An
alternative is EIA (Electronics Industry Association of America) cutoff wavelength,
which states that the cutoff wavelength is:
“The wavelength at which the power in the
HE21 mode is 10% of the power in the HE11
(fundamental mode)”
Since the EIA cutoff wavelength can be 100 nm less than the theoretical cutoff
wavelength it is possible that singlemode operation defined as above could still take
place at 1320 nm. To determine this the power in both the HE11 and the HE21 mode
would need to be calculated.
[13 Marks]
Solutions
2 (a) Material dispersion is pulse broadening in an optical fibre resulting from the different
group velocities of the various spectral components at different wavelengths launched
into the fibre by a source. It occurs when the velocity of a plane wave propagating in
the dielectric medium varies non-linearly with wavelength. This is a result of a
refractive index which varies non-linearly with wavelength, and since the velocity of
propagation is a function of refractive index, the result is a non-linear variation of
velocity with wavelength.
The factors which influence the level of material dispersion are:
1.
The fibre span (this factor is inevitable, since the trend is toward ever
increasing operating spans).
2.
The spectral width of the optical source in use
3.
Fibre doping and thus the operating wavelength of the system and its proximity
to a wavelength at which material dispersion is a minimum.
Material dispersion can be reduced by:
1. Using a singlemode laser with a narrow spectral width, e.g. a Distributed Feedback
laser, (DFB) which will typically have a linewidth of about 10 - 30 MHz.
2. Operating at a wavelength at which material dispersion reaches a minimum.
[6 Marks]
2 (b) To determine the fibre material dispersion parameter D, we recall that
D

c
d n12
d 2
thus:
D  Y.
1
c
Thus in our case where Y is .01 and the wavelength is 1550 nm the value of D is given
by 2.16 x 10-5, with units of seconds/meter.meter. We require an answer in ps/nm.km,
which can be derived by converting seconds to ps (multiply by 1E12), meters to nm
(divide by 1E9) and meters to km (multiply by 1E3), to yield the correct answer of 21.6
ps/nm.km.
For material dispersion the impulse broadening is given by the product of the material
dispersion factor, the source spectral width in nm and the fibre span in km. In our case
we know that the pulse broadening is 3 ns (or more correctly in this case 3000 ps) and
we require the span in km at which this value of broadening will occur. This is clearly
simply:
3000 ps
 44.6 km
spectral width (nm) . D (ps / nm.km)
[8 Marks]
Solutions
2 (c) Equipment:




Honeywell HFE4080-32X VCSEL Laser mounted in an ST package on a laser heatsink
ILX 3412 precision Laser diode Driver and associated mains supply and laser interconnect
lead
Interconnecting ST connectorised 62.5/125 µm fibre patchcord
Fiber Optic power level meter, Megger OTP 620.
Connect the laser to the ILX laser driver (the front cover is shown below) and the laser
optical output to the optical power meter. The laser driver output can be controlled to
within 0.1 mA. Note that to protect the laser the current output is initially off when the
mains power switch is turned on.
Turn on the laser driver using the power button. Turn the laser driver output current
control to zero (fully anti-clockwise) prior to turning on the laser current. To turn on
the laser current press the small output button once (below the current set knob).
Rotating the laser output knob clockwise will increase the current, the value of
which is shown on the display. If the display does not show an increase in current
recheck your connections.
To switch off the laser current depress output button once more. To protect the
laser from transient damage do not disconnect the laser from the laser driver at any
time when the laser current is enabled. The correct sequence to disassemble the
experiment is to turn off the laser output, then disconnect the laser from the ILX laser
driver, then turn off the ILX driver using the power button.
Measure the laser diode light-current characteristic, by varying the laser diode current
from about 1 mA up to a maximum of 12 mA, in small increments (typically 0.2 mA,
but close to threshold smaller increments of 0.1 mA will be needed for accuracy.
Monitor the output of the laser via the supplied ST connectorised 62.5/125 µm optical
fibre patchcord, connected to the optical power meter. Set the optical power meter
to measure µW at 850 nm. WARNING: The maximum laser diode current must never
exceed 12 mA otherwise permanent damage to the laser will result. Plot the laser diode
light-current curve.
The laser threshold is the point where the laser changes its operating mode from a
spontaneous emission (like a light emitting diode (LED)) to stimulated emission
(Lasing). By convention the threshold can be found by drawing a line parallel to
the characteristic above threshold. The point where the line intercepts the X or Current
axis is the threshold.
Solutions
The differential slope efficiency (dL/dI) is the slope of the characteristic at a particular
current. The SE has units of µW per mA (or mW per mA depending on the laser
optical power). The differential slope efficiency can be found by calculating the slope
at a number of sections along the characteristic. E.g. above threshold identify a linear
section and picking two points at the extremes of the sections calculate the slope using.
y 2  y1
x2  x1
[11 Marks]