Idealised Laser Models

In this section a brief description of the idealised three- and four-level laser models are given so that they can be compared to the quasi-three-level model. This discussion of idealised laser models is equivalent to that of (Fan, 1995).

7.3.1 Four-Level Lasers

Consider the energy-level scheme in Figure 7.1 which illustrates the operation of an optically pumped four-level laser.

W

^U

N,

T-4

M3 v

r31 / r32l

/^•±

W

³¹

N,

Figure 7.1: A schematic diagram of an idealised four-level laser model (Fan, 1995). See text for a description of all the symbols.

Initially all the atoms of the gain medium are in the ground level 1. The pump radiation excites the atoms from the ground level into level 4. The transition time (743) is assumed to be infinitely fast so that the atoms in level 4 decay rapidly into the upper laser level 3. This is a non-radiative transition in which the energy difference between the levels is transferred to the crystal in the form of heat. Due to the infinitely fast decay time, the population density of level 4 is assumed to be zero. The laser action occurs when the atoms are transferred from level 3 into level 2 via the process of stimulated emission. In addition to the stimulated emission process, the atoms can also decay from level 3 into level 2 by spontaneous emission. From level 2, the transition time (r2i) to the ground level 1 is again assumed to be infinitely fast so that level 2 has

a population density of zero. In this non-radiative transition, the energy difference between level 2 and 1 is transferred to the crystal in the form of heat. The relaxation from level 3 directly into the ground level 1 is usually associated with fluoresence.

Since levels 4 and 2 have population densities of zero, the system effectively reduces to a two-level model which only accounts for the population densities in levels 3 and 2. Due to the fact that level 2 is assumed to be empty in the four-level laser model, there exists a population inversion as soon as level 3 is populated.

The rate of change in the population inversion density between the upper and lower laser levels can be explained by the following rate-equation (Fan, 1995)

d

^L =

^WuNl

_ ^L _ 2W

³²

AN (7.4)

at

T3

where AiV — N3 — N2 = N3 and Wij is a rate constant given by

where the ij subscript denotes a transition from level i to j . Iij is the intensity of the light that has a photon energy of huij and a is the spectroscopic cross-section.

Note that the spontaneous emission processes out of the upper laser level 3 is denoted by r3 which accounts for all the radiative and non-radiative spontaneous tran- sitions out of the upper laser level.

The time dependence of the laser light intensity in the laser cavity can be described by (Fan, 1995)

^ f = cW32ANhu32 - !™ (7.6)

at TC

where c is the speed of light and rc is the cavity-lifetime denoted by

r

< = S

⁽

">

with L the optical length of the cavity and 8 the logarithmic roundtrip cavity losses that includes the loss of the output coupler. The first term in equation 7.7 represents the roundtrip gain in the resonator while the second term denotes the roundtrip loss in the resonator.

The most widely-used solid-state laser, namely the 1064 nm Nd:YAG laser is a very good example of a four-level laser.

7.3.2 Three-Level Lasers

The main difference between the three- and four-level laser models is the fact that the ground level is the lower laser level in the case of the three-level model. Figure 7.2 shows a schematic diagram which illustrates the operation of a three-level laser.

W

^l3

N

^t

W

^1X

{N

²

-N

^X

)

Figure 7.2: A schematic diagram of an idealised three-level laser model (Fan, 1995).

See text for a description of all the symbols.

To begin with, the atoms are in the ground level 1. The incident pump radiation causes the atoms to be excited into level 3. Similarly to the four-level laser the non- radiative transition between the upper pump level and the upper laser level is assumed to be infinitely fast so that the population density in level 3 is zero. The assumption of an infinitely fast transition between level 3 and 2 effectively reduces the model to a two level model. The laser action transfers atoms from level 2 into level 1 with the process of stimulated emission. Since the lower laser level is in fact the ground level, it is possible for an atom in the ground level to absorb a laser photon and be excited to level 2 again. The reabsorption of the laser light introduces an additional loss in the system. Due to the fact that there are essentially only two energy levels, more than half of the atoms have to be pumped into the upper laser level 2 in order to reach a population inversion. This causes that more atoms to contribute to the non-radiative transition between levels 3 and 2 which adds heat to the crystal. Even more significant is that the populated lower laser level results in a higher laser threshold than in the case of a four-level laser. The rate-equation that describes the population inversion in the three-level laser model is given by (Fan, 1995)

dAN NT + AN

a A i Vdt T = 2W13N1 - T + - AW2 2l AN where AN = N2 - JVi and NT = Ni + N2.

(7.8)

The additional factor of two that is present in the pump and stimulated emission terms is due to the system that comprises of the upper and lower laser level. This implies that an atom that is pumped from the ground level 1 into the upper laser level 2, changes the population inversion by two. The same argument is true for the stimulated emission laser action which transfers atoms from the upper laser level into the ground level. The transient behaviour of the laser light intensity is written in analogy to the four-level laser as (Fan, 1995)

^ = cW21ANhv21 - — (7.9)

at TC

Due to the reabsorption loss in three-level lasers, they are less efficient and require higher pump densities than four-level lasers. A good example of a laser which operates according to the three-level model is a ruby laser.