The basic idea underpinning an optical fibre laser and an amplifier is relatively simple.
A diode laser provides the optical power in order to excite the rare-earth ions, the cavity
is usually made by two mirrors coated in order to obtain high reflection at the lasing
wavelength, which confines the light and provides the optical feedback necessary in
w avelength, w hich is filtered out after the m irror output coupler, the layout is show n in figure 2.7 A. O ther solutions that have been im plem ented are based on fibre Bragg - gratings [29] and fibre loops [30], instead o f mirrors.
D oped Fibre
A
MM"
Lens
Laser Dkxle (Pump) Mirror O utput C oupler Mirror
B
L aser DkxJe (Pump)
D oped Fibre
Figure 2.7: Upper, diagram A shows a Fabry-Perot fibre laser while diagram B is a fibre amplifier.
In a rare-earth doped fibre am plifier the m irrors are rem oved, and the pum p is needed to establish population inversion in the fibre active medium. The dopant usually em its fluorescence at w avelengths w hich are located in the telecom m unication band betw een
1.3 |0,m up to 1.6 ^im; an incom ing signal can be am plified by stim ulated em ission w ithin the am plification bandwidth. This guarantees the coherence o f the photons em itted w ith the signal. U sually, isolators are placed at the end o f the active fibre, reducing the im pact o f back reflection. Low loss-connectors (FC type) are anchored on the front end in order to allow connection w ith optical cables for the input and the output o f the signal (fig. 2.7 B).
T he gain in the am plifier is defined as [31]:
w here P,„ and Pout are the am plifier input and output signal pow er and Pase is the noise pow er generated by the am plifier, w hich lies w ithin the optical bandw idth o f the m easurem ents. In m ost cases, Pase is very low com pared to the signal and can be neglected, m oreover it is com m on practice to define the gain in dB as:
(2.18) G ,,= 1 0 1 o g
^ p ^
oul P \ m y
Predicting the gain is complicated by the local changes in power and signal that can affect the population of the excited states. At any given point in the fibre, it is possible to define the so-called gain coefficient as:
(2.19) g iz ) = (7^,N^-cT^N,,
where the is the stimulated emission cross-section and cr^ is the absorption cross -
section experienced by the amplifying signal whilst the Nu and N j are respectively the
density of ions in the upper and lower levels involved in a transition.
The net gain G can be integrated if the explicit relation of the density with the fibre length is known, as:
(
2
.20
) or: (2.21) G = e° I G j g = 4 3 j g i z ) d z .2.11
A nalytical M odel
The behaviour of rare earth ions in doped fibre lasers and amplifiers can be modelled by a phenomenological model which takes into account measured parameters of the
transitions involved as: cross-sections and decay rates. The time dependent population
densities of the excited state are subject to a set o f differential equations called rate-
equations. If the dynamics of the system are not under investigation it is usually possible to calculate directly the steady state value of the population densities without having to integrate a time dependent system of first order differential equations. It is clear that when the system is in the steady state, the rate o f change of the population is
zero and the rate-equations are reduced to a linear equations system. M oreover the
system has to include the conservation law:
(2.22)
N,,,=Y^N;,
i
where Ni are the population densities of each j-excited level and N,ot is the total doping
The dopant is usually expressed in percentage by weight, this is sim ply the m ass o f the dopant added during the preparation o f the preform divided by the total m ass o f the glass and m ultiplied by 100. Sim ilarly in ppm w (part per m illion by w eigh t) is the previous fraction m ultiplied by 10^:
ffl
(2 .2 3 ) c ( p p m w ) = — ^ 1 0 ^ .
^ Z B I A
O ne m ole o f Z B L A w eighs (sum o f all product o f the molar fraction and m olecular w eight o f the fluorides com ponents or W) 167.8 gr. so 0.335 gr. o f dopant in Z B L A is needed for 2 0 0 0 p.p.m .-w doped fibre. If the density o f the glass S is know n, the density o f ions can found for thulium ( P M d - 6 9 ) as:
ions J. / 5x iU' “c
PM^ cm^
•j
In the literature the density is reported to be 4.3 gr/cm [32]. If c is express in p.p.m - m ol the density o f ions is:
ions
(2.24)
w
cm^
Optical quantities such as pump pow er usually depend on the position along the fibre and they fo llo w a first order differential propagation equation. S ince these quantities appear in both the equations system s, in w hich they are coupled, a sim ple analytical solution is rarely p ossible and numerical m ethods must therefore be applied.
The full rate-equation numerical description is postponed until chapter 4, w h ilst here a sim plified analytical solution w ill be given . An approximated analytical solution is essential in order to obtain a numerical solution. Indeed, the numerical integration m ethod relies on a set o f guess solutions, w hich w ill be iteratively relaxed until a converged solution is found.
In this section, the system form ed by the four low est lev els o f thulium ions w ill be analysed. This is the building block for the more com plex m odels presented in the rest o f the thesis. The energy lev els are show n in fig. 2.8. It is p ossib le to ignore a b initio
the population o f the level, this has a sm all impact in the calculation o f the
population densities, the error o f the order o f the decay branching ratio (1 %) for the 2.3 |0.m transition, m oreover the energy gap with the low er lying energy lev el ^p4
is comparable to the phonon energy of the zirconium glass. This creates a fast depletion by non-radiative relaxation and in the steady state, a population of zero is quite suitable.
X I Q i
1.5[ _
E
^ 1 >- uc
0) 3 CT (U.“r 0.5
E
^ 2 1 > ro 0 Cnrv
0
> 0Nj ^H
4( i msec)
^^4(10 msec)
0 " 6Figure 2.8: Lower energy Thulium levels used in order to give and approximated
analytical solution.
The density populations N, o f the levels are filled according to the following set of first order differential equations;
dt dt dN dt
Nro,=jL^.
where cr^ is the absorption cross-section for the H6-> H4 transition, and <I> i s the flux
pump rate, i.e. number of photons per unit o f time and area and the are the transition
rates from the level i to the level j.
Steady state conditions require that the time derivative is zero. It is useful to define two constants in such a fashion that they have the property of being unit-less for the first and for the second, a rate per unit area, which gives the followed equations: