Er 3+ Rate Equations

The transitions of electrons between the different energy levels of the Er³⁺ ions can be mathematically represented by rate equations. These equations will describe the governing mechanisms behind population changes during stimulated absorption, stimulated emission and spontaneous emission with respect to traversing photons from an incident field of light.

The transitions of electrons between the energy levels of the Er³⁺ ion can be represented by the three-level system as shown in Figure 27. The three energy levels are designated as 1, 2 and 3 with 1 being the ground state, 2 being the metastable state and 3 being the excited state. The transitions between the states are either absorptive or emissive in nature, and are represented by the notations , and for the

27 The reduction in performance is only at the initial point of lasing or amplification, before stimulated emission begins. As such, spontaneous emission occurs primarily when the external signal is absent, and once the lasing or amplification of a signal begins, the negative effects of spontaneous emissions are rarely seen. However, this is only valid for the case of CW systems. Pulse systems are highly susceptible to the effects of spontaneous emission, as the pulse creates multiple initial states, and can see the performance of the amplifier being affected.

77 pumping rates, stimulated emission rates and spontaneous decay rates respectively.

The superscripts and represent radiative and non-radiative transitions, while the subscripts 1, 2 and 3 each represent the transiting energy level, with the first subscript denoting the originating energy level, and the following subscript denoting the destination energy level.

Figure 27: Energy level diagram corresponding to the pumping rates. Radiative and absorptive transitions are denoted by solid lines, while the dashed lines denotes non-radiate transitions

From the figure, it can be seen that nine transitions are possible. Pumping rate transitions take place between level 1 and 3 directly, bypassing level 2 to give an upward transition of or a downward transition of . Spontaneous decay can also occur directly between levels 1 and 3 and are radiative in nature. This transition is represented by in Figure 27. Between levels 3 and 2 there are two possible transitions, namely , which is non-radiative in nature and , which spontaneously emits as it decays. Between levels 1 and 2, four possible transitions can occur. Two of these transitions are stimulated emissions, represented by and , while the other two occur spontaneously, and are represented by and .

In order to model the rate equations of the Er³⁺ ions, it is assumed that the spontaneous decay from level 3 is predominantly non-radiative, and is given by:

≥ ………..……..…...…………..(9)

3 (Pump Level)

2 (Metastable Level)

1 (Ground Level) Energy

78 where

= + …….………….….….….……(10)

In the same manner, the decay from the metastable level to the ground level is a combination of radiative and non-radiative decays, and is defined as:

= + ….….……….….……(11)

whereby:

= ………..……...….…….(12)

with being the fluorescence lifetime. The spontaneous decay from level 2 to 1 is predominately radiative, thus giving:

≥ ………….…………...….……(13)

The populations at levels 1, 2 and 3 can be denoted as , and . The atomic rate equations corresponding to these populations can be written as:

= − + − + + …...(14)

= − − + …………....…..….. (15)

= – – ………....……....(16)

The time derivatives in a steady-state situation are zero, thus giving:

79

= = = 0………..………...…..…… (17)

As the transitions between the excited state and metastable states are very fast due to the high probability of these transitions occurring, thus is almost always very close to zero, with most of the population remaining at levels 1 and 2. In this regard, Equations 14 and 15 can be solved as follows:

= ………. (18)

= ………..…………(19)

≈ 0……….…………..…….…..………(20)

with R being taken as [63]. These population densities at each level are given as [67]:

= + − − + ′ ………(21)

= − − + ………...…....(22)

= − + + ′ …………...………...…(23)

+ + = …………...……….….…..…….(24)

80 whereby is the total dopant concentration. Exposure of the Er³⁺ ions to incident light at 980 nm will result in multiple transitions occurring between the three levels.

While it is possible to accurately represent the actual transitions taking place in a mathematical model, it is also very complex. However, certain conditions can be taken into account to build a model that is still accurate, albeit much less difficult. By taking into account the fact that transitions from the ⁴I11/2 to ⁴I13/2 levels are predominately non-radiative and those between the ⁴I13/2 and ⁴I15/2 levels are radiative in nature, equations 21 to 24 can now be simplified into just two equations:

≫ + ………...…….……… (25)

≫ ……….………...(26)

As is a very fast non-radiative transition, it can thus be assumed that the transition rate of is much higher than the pump or stimulated emission rates, such that

≫ _, . Therefore, the complex three-level system can now be essentially represented by the simpler two-level model.

As the time derivatives vanish in steady state conditions, the distribution of the excited electrons in the excited state is essentially a Boltzmann distribution. In this regard, the populations at the ground and excited levels can now be described as:

= ^{( )}

≡ ……….……...……….……(27)

where

= ^{( )}……….…..……...…….(28)

with being the energy at the excited state, and being the energy at the metastable state, and represent the Boltzmann constant and thermodynamic

81 temperature of the group of atoms, respectively. The inversion level, can now be written as:

= −

= ^{( )}

( ) ……...…..…..……….……....(29) where and are the pump and signal wavelength powers. The lifetime of the metastable state is given as .

Equation 29 provides two crucial insights into the transition mechanisms that take place in the Er³⁺ ions when excited, which in turn explain the behavior of the EDFA. Firstly, it is plain to see that the inversion levels are heavily dependent on the pump and signal wavelength powers. Secondly, the factor shows that pumping at 980 nm is more efficient than pumping at 1480 nm. This is because when excited by 980 nm pumping, the electrons orbiting the Er³⁺ are raised to the excited state level, but quickly decay to the metastable level as a result of the small lifetimes experienced at this level. The difference in terms of energy levels between the excited and metastable states is approximately 0.4 eV. Taking this energy difference into account, the obtained value of becomes about 0.0. However, pumping the Er³⁺ ions at 1480 nm on the other hand cause thermalization²⁸ of the excited ions to occur at the within the metastable level itself and in turn increasing the value of value to 0.4 [63]. The increase in the value of can be seen in the inversion factor. When a strong pump power is introduced to an EDF, along with a small signal power such that W τ ≫ 1 and W ~ 0, pumping at 980 nm gives an inversion factor of almost 1. On the other hand, pumping at 1480 nm gives an inversion factor of 1.6, less efficient than that obtained at 980 nm pumping.

28 Thermalization is used in place of non-radiative transitions as the energy released during the relaxation of the excited electrons is in the form of heat. In reality, thermalization and non-radiative relaxation transitions are one and the same.

82 As the optical properties of the Er³⁺ ions are essentially a function of the transitions between the various energy levels, thus the rate equations form the building blocks from which the subsequent behavior of the Er³⁺ ions can be understood. However, it must be taken into account that the factors that govern the behavior of the EDFA are not limited to those of the Er³⁺ ions specifically; even the host material can have a significant effect on the performance of the EDFA, particularly in terms of the absorption and emission cross sections. This will be described in detail in the next section.