Power of the pump at the lasing threshold
Output mirror transmission
The output mirror transmission should be chosen according to the gain available in the amplifying medium. As stated in the lesson, the product of the gains in both directions G+G- must be more than 1/R1R2 (see Figure E4 for the sizes) so that laser oscillation can occur. Here, the mirror coated on the Nd:YAG crystal is assumed to be very reflective so R2=1. However, energy loss can occur in the cavity at the crystal-air interface or by diffusion via dust on the mirrors. By convention, these losses are taken into account by giving the second mirror M2 a reflection coefficient slightly less than 100%. Generally, these so-called passive losses make up 1 to 2% in this type of laser cavity so R2 is about 98%. As the output mirror transmission of M1 is very small (T1=10%), the intensity of the laser will not change much before and after the crystal so it can be assumed that G+=G-.
As R1=1-T1, the condition for oscillation is: G2>1/R1R2.
When the pump is at full power and for a signal at 1064 nm, G02 is equal to 2.25 according to the order of magnitude given in the section “The amplifying medium pumped by a diode”. The fraction 1/R1R2 is equal to 1.13 thus the the lasing threshold is exceeded.
Power of the pump at the lasing threshold
It is possible to calculate the power of the pump Pp necessary to reach the oscillation or lasing threshold (G02=1/R1R2). G0 must be given as a function of the power of the pump by using the formula from the section“The amplifying medium pumped by a diode”:
where
Thus
so the pump power at the threshold is 77 mW.
Why are the laser beams so small ?
The beams at 808 nm and 1064 nm have a radius of about
inside the crystal. This may seem very small but it is necessary so that the number of ions per unit of volume is sufficient and also so that the number of photons at 1064 nm is enough to trigger an efficient stimulated emission. The formula that calculates the gain according to the irradiance can be modified to include the power of the pump and the radius of the pump beam,
. Assuming that the beams at 808 and 1064 nm have the same radius, it is possible to calculate the minimum radius needed to reach the oscillation threshold with an output mirror transmitting 10% and a maximum pump power.
Using the formula
with the following conditions given in the section“The amplifying medium pumped by a diode”:
for a pump power of
focused in the crystal with a radius of
,
then
To be at the threshold with the maximum pump power,
must be equal to 1/(1-T)
where
then the value of the radius r is calculated as
. This means that if the diameter of the beams is more than this value, then the power of the pump is not high enough to reach the oscillation threshold. The beam radius must therefore be much less than a millimetre.