Notes ... Part 4

PH3310 Lasers and Modern Optics

2.3 The Lineshape Function

So far we have assumed that the atomic energy levels are infinitely sharp.

However, transitions emitted by real atoms possess a lineshape function since in real life there exists some uncertainty about the exact values of and , due to both intrinsic and extrinsic causes.
Define as the probability that a stimulated photon will have a frequency in the range

The precise form of depends on which of a series of possible broadening mechanisms dominates the situation.
i.e there exists various contributors to depending on the atom's environment.

The narrowest linewidths are obtained when
-- atoms are at low T -- close to being stationary
-- atoms are far apart -- dilute gas
-- radiation density is low.
In this case produces a broadening known as

(i) Natural broadening -- which is a quantum mechanical limit due to the natural lifetime of the level(s) producing the transition.

where  sec (typically) There exists radiation damping Lorentz profile

If the T of the gas increases the linewidth increases due to:
(ii) Doppler broadening -- due to thermal motion of emitting atoms.

where v is the component of the velocity of the atom in the direction of the line of sight.

Maxwellian velocity distribution of velocities thus bell-shaped distribution of .

since 1-D.

Thus

e.g. at 400 K 52 MHz GHz.

If the gas pressure increases then the linewidth increases due to:
(iii) Pressure (or Collisional) broadening -- if atom suffers a collision whilst in an excited state -- may induce a downward transition --
i.e. collisional de-excitation.

where is the spontaneous decay rate and is the collision induced rate.

In the cases of high powered radiation beams there exists
Power broadening due to -fields of the EM waves perturbing the atoms Stark broadening of ions at high densities.

Thus the previous argument needs slight modification:

A technical Aside and some practical advice:
Line broadening mechanisms divide into two main types termed:
homogeneous -- when each atom has the same lineshape function characterised by Lorentz profile, natural and collisional.
and inhomogeneous -- characterised by Gaussian profile, Doppler broadened when comprised of an ensemble of different groups of atoms each with a different lineshape function .

For homogeneous broadening:

therefore if

For inhomogeneous broadening:

therefore if   

NOTE:
Thus it will be sufficient for our purposes to approximate either of the above expressions by when we wish to incorporate the use of the lineshape function.

2.4 Population Inversion schemes.

Take a first look at the question of how to achieve a Population Inversion

? 2-level system

We shall see that even with very intense optical pumping the populations of levels 1 and 2 can at best only be made equal. (since )

? 3-level system          like the first laser -- a ruby laser.

NB. Broad band is desirable for more efficient pumping.
We shall see that 3-level systems are inherently inefficient -- they require very high pumping powers since > 1/2 of the groundstate atoms have to be pumped to the upper level to achieve inversion. Therefore this energy is wasted.

? 4-level system -- whereby pumping requirements are greatly reduced.

Thus it is possible to obtain inversion between levels 1 and 2 whilst there still exists a large population in the groundstate level .

We shall return to this topic for a more detailed consideration later.

Progress Quiz 0:

1. Define what is meant by stimulated emission and spontaneous emission and mention how the resulting photons may be characterised.
2. Describe what is meant by a laser medium.
3. Why will a laser only work when more of its atoms or ions are in the upper laser state than in the lower laser state?
4. If you were to measure the electric field in the light from a searchlight you would find that it fluctuates randomly -- why is this so?
5. Describe what is meant by coherent light.

Assignment: Read sections 1.5, 1.6 and 1.9 of Wilson and Hawkes.