As with electronic oscillators, there are many different types of laser. Here we can consider a sort of ‘basic’ laser arrangement as illustrated in figure 4.2. For the sake of simplicity we'll ignore many of the details and effects which arise in real lasers. All we are concerned with here is the basic process by which oscillation takes place. The illustration shows a length, l, of a suitable gain medium placed in between a pair of refocusing mirrors which define a resonant cavity. In practice, one of the mirrors will usually be built to be semi-reflecting or ‘leaky’ to allow some of the resonant oscillation field to escape as the familiar laser beam.
In this case, the system's gain comes from a population inversion produced by pumping the medium in some way — e.g. with a high power light source. This pump provides the power input needed to drive the oscillator & performs the same function as the electric power lines in a conventional electronic oscillator. The operation of the device can be understood by considering a small field oscillation
at the angular frequency, 
, which starts off just beside the low-loss mirror, 
, and moves towards the ‘leaky’ mirror, 
.
After passing through the gain medium the field emerges as
where 
is the single pass field gain of the medium at the frequency, f. (Note, calling it a gain may be optimistic. If 
the field emerges smaller than it began and the ‘gain’ is really a loss!)
This produces a reflected amount
where 
is the field reflectivity of the leaky mirror. The reflected field passes back through the gain medium to re-emerge as
which reflects off the mirror whose field reflectance is 
a field
The field has now completed one round trip back and forth along the laser cavity. In effect, the field has gone ‘once around the loop’. In this case we can identify the loop gain as being
As before, oscillation is possible only if this gain satisfies the Barkhausen Criterion at one or more frequencies. The basic properties of a typical laser's loop gain are illustrated in figure 4.3. This shows how the modulus and phase angle of the loop gain vary with frequency.
In general, light passing through the gain medium will tend to be attenuated. Similarly, the reflectivities of the mirrors will always be such that 
. As a result, we find that 
is less than unity at most frequencies. However, the stimulated emission mechanism can cause 
over a small band of frequencies centred on one of the gain medium's emission lines. Oscillation is therefore only possible inside such a limited band surrounding the nominal line frequency.
For oscillation to occur the phase delay for a round trip must be a whole number of wavelengths. This means that the oscillation field's wavelength, 
, must obey the condition that
where n is an integer, l is the cavity length, and m is the refractive index of the gain medium. Since 
(c being the speed of light in vacuo) this is equivalent to saying that oscillation can only take place at frequencies
These expressions may be used to identify any oscillation frequencies which satisfy this requirement and fall inside the band where the system has gain.
In many situations we can assume that the refractive index of a material has some frequency independent value, 
. However, this assumption may not be acceptable when we are examining the behaviour of the optical oscillator considered here. This is because the index of a material tends to be strongly frequency dependent in the region near an emission line. This behaviour is illustrated in figure 4.3. The dashed line shows the phase/frequency behaviour assuming a steady refractive index. The full line shows the phase/frequency behaviour with the line effect taken into account. It can be seen that this alters the frequencies at which oscillation is possible. To take this into account we should rewrite 4.15 as
to explicitly recognise this effect. The situation illustrated in 4.3 has been deliberately chosen to show one of the possible consequences of this line dispersion effect. Looking at figure 4.3 we can see that, in this case, oscillation is possible at two frequencies where the round trip is either n or 
wavelength long. From the plots it can be seen that 
. This situation would be impossible if the refractive index were frequency independent. The above analysis is deliberately a very simplified one. Many other effects (beam diffraction, etc) should be taken into account when we wish to determine whether a given system can oscillate, and — if it can — at what frequencies. It does, however, serve to make the basic point that the conventional electronic feedback oscillator and the cavity laser share the same basic features.
Summary.
You should now know that feedback oscillators require a system which combines some gain mechanism with a feedback arrangement. That oscillation requires two conditions to be satisfied simultaneously:
- the loop gain must have a magnitude greater or equal to unity;
- the phase delay around the loop must be an integer number of cycles.
It should also be clear that the oscillation power power must be put into the system in some way — e.g. by electric power lines, or an optical pumping mechanism. It should be obvious that the output oscillation power can't exceed this input level.
Content and pages maintained by: Jim Lesurf (jcgl@st-and.ac.uk)
using
TechWriter Pro and HTMLEdit on a StrongARM powered RISCOS machine.
University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.
