The Population Inversion

It was stated earlier that the population inversion required for net gain to be observed in any laser can be generated through the introduction of additional energy levels to those of the lasing transition. Consider the four level energy diagram shown in Fig. 2.3.

Pump band

Pump

Transition TraLasensitionr

Ground state

Figure 2.3: Simplified energy level diagram for a four level system.

Essentially pump energy is supplied to the system which is of an appropriate frequency to excite the electron from the ground state into the broad pump band. From here rapid non-radiative decay to the upper laser transition level takes place through phonon interactions with the crystal lattice. It is normal that this is a long-lived or

metastable state from where a radiative transition to the lower laser level takes place through either the process of spontaneous or stimulated emission. The electron then undergoes a further rapid non-radiative decay returning it to the ground state. Given then

a sufficiently high pump rate and suitably timed decay rates, specifically ^ ^32

ideally also x < x^^ (spontaneous), then a population inversion can be achieved.

Additional decay processes are also noted in Fig. 2.3, namely; x^^, x^^ and x^^.

These may be regarded as loss mechanisms, reducing the number of electrons making the

2 ^ 1 laser transition. Fortunately, the frequencies of these transitions are too high to take place via single phonon interactions and consequently rely on either photon or multi- phonon processes. The decay times for such processes are generally at least three orders of magnitude longer than the single phonon interaction, ensuring small relaxation rates compared to that of the non-radiative decay to the upper laser level.

When evaluating the size of the population inversion it is necessaiy to include in the lower laser level population not only that population due to the laser transition but also any thermal population due to its proximity to the ground state. This equilibrium population is described in the usual way by the Boltzmann distribution

N , _ f - A E

No kT (2.24)

To place this in context, in a gain medium such as Neodymium doped Yttrium Aluminium Garnet (NdiYAG) the lower laser level for the I pm laser transition is around 2000 cm 'l above the ground state. At room temperature equ. (2.24) then gives a population ratio of 7x10'^! In contrast, however, for the three level mby laser the lower laser level is the ground state and consequently has a population near unity.

In the regime then where the equilibrium population of the lower laser level is very small and the level has a fast decay time, it becomes possible to create a population inversion even as the pump power approaches zero. However, as this lower state

approaches the ground state thermal repopulation becomes a significant factor. In the

extreme, lasers exist which operate on a three level basis where the ground and lower laser transition levels are the same state. In this instance it clearly becomes necessary to excite around one half the atoms from the ground state to create the population inversion.

Chapter 2: Background LASER Theory

Consequently such lasers are generally more difficult to operate, requiring considerably higher pump powers to attain threshold.

2,2.5 The Laser Rate Equations and Operation at Threshold

In the preceding sections the basic building blocks to laser theory have been discussed, including; spontaneous and induced electron transitions, transition broadening effects and the 3 or 4 state energy level diagram which provides a means of creating population inversions and consequently gain in a laser media. It is at this point that these concepts are now brought together to begin to form a model describing in practical terms the operation of lasers. In particular, the optical pumping of four level, rare earth ion, solid-state gain media is considered.

The gain experienced by a monochromatic radiation field as it transits the laser medium can be derived from the rate equations describing the populations of the upper and lower laser transition levels. These are;

^«2 dt 8 2₈₁ n c(j)a + — -Wj,n^ , (2.25a) T21 dn^ dt C0CT + —_^21 , (2.25b) ^tot =n-^+n2 + riQ . (2.25c)

Equations (2.25a,b) essentially bring together the spontaneous and stimulated transition rates described previously with the final parameter in each describing now also the pump rate and lower laser level decay rate for the four level system. Some reorganising of the stimulated transition rate has been done in order to express the B^^p(D) product in terms the more useful stimulated emission cross-section and photon

flux parameters, namely

B21 = ■; ^ ; (?2i(D) and p(D) = hDg(p)()) , (2.26)

hDg(D) so that

B2iP(^) — CG2|(D)(|) . (2.27) It is to be noted that in these expressions for the relative populations of each level, equ. (2.25), certain assumptions have been made: Firstly, that the pump transition is

rapid and secondly that the nonradiative decay time from the pump band to the upper laser

level is short. Consequently, the total number of atoms involved in the system is just the sum of those in the laser transition levels and the ground state as described by equ. (2.25c). Making the further assumption that the lower laser level population is zero, this being valid for when is small, gives the laser rate equation for the population inversion as

^ = - n 02|( |) c - i + W /n ,„ ,- n ) , (2.28)

dt Xf

where x^ is the effective fluorescence lifetime of the upper laser level, taking into account

all spontaneous decay transitions, and is the effective pump rate, taking into account

all spontaneous transitions from the pump band to energy levels other than the upper laser level.

Rather than looking at the time rate of change of population inversion, it can be useful to consider the time rate of change of photon density within the laser. This can by analogy to equ. (2.28) be seen to be given by

^ = n ca2|( |) - - - ^ + S . (2.29)

dt r X,

The change in photon number due to stimulated transitions is of course identical to the change in population inversion due to stimulated transitions, as is seen in the first factor on the right of equ's. (2.28) and (2.29), The factor 1/1', where 1 is the gain length and 1’ the resonator length, must however be included as (j) is a photon flux and the average flux in the resonator increases at a rate 1/1' slower. The second factor in equ. (2.29) describes the reduction in the intra-cavity photon density due to cavity losses such as the reduced miiTor reflectivity of a laser output coupler, parasitic losses from optical surfaces and absorption losses. For completeness an additional factor describing the contribution

Chapter 2: Background LASER Theory

smaller than either the stimulated emission or cavity loss components and so is henceforth neglected.

Under the conditions of either steady-state operation or radiation field growth, the time rate of change of photon density is respectively equal to or greater than zero. From equ. (2.29) one can then say that the required population inversion condition at threshold is

It has been shown earlier that is inversely proportional to the transition linewidth. Thus, in order to obtain a low lasing threshold a narrow transition linewidth and long cavity decay time x are desirable, noting that the cavity decay time is determined by a combination of output coupling and parasitic losses. It is clear then that as one would expect losses should be kept to a minimum. However, increasing the output coupling reflectivity, while reducing threshold, will reduce the fraction of circulating

power coupled out of the laser cavity and thus some trade off must be made to obtain the

most efficient combination.

It can be useful to express the threshold conditions for operation described above in more measurable quantities. Consider the laser resonator depicted in Fig. 2.1, consisting of two mirrors of reflectivity and R^, separated by some optical distance 1' containing

the gain medium of length 1. Threshold is reached when the photon density after one

round trip of the cavity just equals the photon density at the start of the round trip. For

then a system in which the single pass gain of the medium is described by exp(g^l), at

threshold

R iR 2exp(2gol-2al) = l , (2.31)

where a describes the absorption and scatter losses in the gain medium, and R^ is a modified reflectivity value for the normally highly reflecting M2, taking into account parasitic surface losses. The factor of 2 in the exponentiation function arises in considering a round trip of the resonator and hence double pass of the gain medium. Given the round trip time of the resonator is just 21'/c and the fractional loss in power in each round trip is therefore just 21Vcx^, then at threshold one can rewrite equ. (2.31) as

21’

X, . -

(2.32)

[ln(R,R2)'' + 2al'

All of these factors are either known or reasonable estimates can be made, allowing one

to predict with some accuracy cavity decay times and consequently, with the additional

measurement of the spontaneous transition lineshape and decay time parameters, allow

good estimates of threshold inversion conditions.

It is of interest of course to know how much pump power need be applied to the

gain medium to sustain threshold operation. Under steady-state operating conditions dn/dt is zero and furthermore, at threshold the photon density is very small and can be ignored. Incorporating these assumptions into the four level laser rate equation described by equ. (2.28) gives

Clearly then for the four level system described here any finite pump rate results in an inversion being created in the gain medium. The threshold inversion requires one to take

into account cavity losses however, and so by equating equ. (2.33) to that given

previously in equ. (2.30) one obtains the threshold inversion pump rate as

This expression demonstrates clearly the desire to have a low loss cavity in the

form of a high x^ and a gain medium with large emission cross-section, long lived upper

laser level and high active ion density. It will be discussed later however, how in the context of many solid-state gain media, increasing active ion densities can lead to an increase in threshold inversion due to increased scatter loss (so reducing x^) and a reduction in x^.