Study of Resonator Properties of Stable and Unstable Lasers

In this work, the stability of optical resonator for two types of lasers are studied which are, He-Ne laser and CO2 laser. The first type, is a low power laser and 0.6328 μm of wave length and the second type (CO2 laser) is a high power laser and 10.6 μm of wave length . Theoretical part deals with illustrating the physical meaning of stability factors which are g-factor, Fresnel No. , Q-factor and diffraction losses , also the mathematical equations of these factors and the geometrical specifications related with optical resonator. According to the figures which show the relationships between the stability factors, He-Ne laser is more stable than CO2 laser ; it has been verified and this is due to the high difference of photon s energy and also to the difference in output powers for the two laser.


Introduction
The gain of laser resonator is a measure of the amount of the light intensity increases by the stimulated emission for one round trip inside the resonator (this trip is starting from the output coupler OC thru the active medium, reflected off the high reflector HR back thru the active medium, backing again at the OC).The gain of laser medium is the light intensity increasing due to the stimulated emission from one of two ends of the active medium to the others.There will be lasing if togther with and the total losses(the output laser is one of these losses), is more than one. The output power will be increase till the losses bring down to preciely one (that means the laser blows up),all the losses is due to non-linearties in the lasing process and finite pumping input.The output power decreases and finally damps if the is less than one. In the same side of the output beam, the losses will built up from imperfect mirror (absorption at the OC and the partial reflction at the HR), Absorption with the reflection at the Brewster window as well as the absorption an d scatter in the lasing medium. [1] A perfect He-Ne laser may addionally have of only 1.01 to 1.05 that depends on the length of resonator (1 to 5%). All optics must be close to perfection as viable to get whatever out of a short tube. For these reasons, the following approximate equation for (LRG) can be used: [1] (approximate) = L * G …(1) Where: • L: length of lasing medium (discharge bore, rod, etc.). • G: gain per unit length.
In both cases, the total full trip ( ) will be: = R ( While the is determined whether a given configuration will be a laser or not, the on hand power that can be drawn from every spectral line will have an affect on the real output power from the laser. In different words, where all different factors are equal, a low gain line may also truely produce a higher proportion of the output power than a high gain line at higher power input.[2] [1]

Stability of Resonator
A resonator of laser can be both stable or unstable. This does not now typically refer to a design that will not now be flexed or distorted due to mechanical stress or temperature variations (though that is additionally a exact requirement for a most lasers, unless deliberatelly introduced so that sure parameters like fantastic mirror alignment can be adjusted with the aid of a feedback control system in similar to adaptive optics in high performance telescopes). The design of the resonator is the one which is accountable for the kind and form of laser beam which is produced. A foremost section of the laser beam is a function of the cavity optics (as properly as the length and cross-sectional form of the real bore and different factors).[2] [1] The key equation determines whether a given configuration of mirrors will end result in a steady resonator is: : the curvature radii of mirrors 1 and 2 respectively. • L : the displacment between the two mirrors.

Fresnel number
Essentially, the Fresnel number was added in the context of the diffraction concept for beam propagation. If a light wave first passes thru an aperture of size (e.g. radius (a)) and then propagates over a distance L to a screen, the situation is characterised with the Fresnel number. [3] = …(6) : is the size characteristic (e.g. radius) of the aperture : is the distance of the screen from the aperture : is the incident wavelength.
The value of F is very impotant to determine the type of the diffraction, so there are two special cases: , geometrical optics laws are applied. The idea of the Fresnel number has additionally been utilized to optical resonators (cavities), in specific to laser resonators. where (a) is now the radius of the back mirrors, and L is the length of resonator. [3] [4] The losses of diffraction, at the back mirrors, are small for the typical mode sizes (i.e. not near the stability limit of the resonator, where the mode sizes can diverge) leads to large Fresnel number (F>1) of resonators (cavities). This is the ordinary situation in a stable laser resonator. Conversely, a small Fresnel number means that the diffraction losses can be significantparticularly for higher-order modes, so that the diffraction-limited operation may alsobe favored. [6] Most stable laser resonators have a fairly large Fresnel number, whereas small Fresnel numbers occur in unstable resonators, which are sometimes applied in highpower lasers. [5]

Q Factor and Diffraction Losses in Laser Resonator
The 'Q' factor of a laser resonator is analogous to the Q factor of a tuned circuit. It is a measure of the energy stored in the cavity versus the losses as the light bounces back and forth between the mirrors. Some definitions of the Q factor of a laser resonator are: [6] [7] = …(7) Where: • E : stored energy in the resonator. • δE : lost energy for one trip.
The final equation of Q factor as a function of the wavelength , the length of resonator and the losses of resonator due to the diffraction is given by: = …(8) where ( ) is the diffraction losses which is given by: [6] = − √ − …(9) where: F: Fresnel Number.

Laser type Specifications
He-Ne laser CO2 laser  (1), shows the relation between the g-factor and the resonator length (L). It ' s clear that, the g-factor decreases-in the two lasers-with (L) at the range (10-50) cm. Therefore, the stability of the two lasers is still in the range of equation (4), which means that the stability of the resonator decreases with the increasing of (L) because it depends on the curvature radii of mirrors and the distance between them but not depending on the type of the laser medium.
In figure (2), the resonator length (L) versus the Fresnel Number (F) is plotted. Nonlinear curves show the decreasing of (F) factor with the increasing of (L) within the range (10-50) cm for the two lasers. The illustration of this behavior is that: at a large values of (L), the resonator losses(diffraction, scattering….) will built up that means the (F) will be less than or approach one for CO2 and He-Ne laser respectively, which is also verifying by the figure (3).In the He-Ne laser the range of (F) is much larger than that in CO2 laser ; this comes from the fact that the stability of the former laser is better than the latter The relationship between Fresnel Number (F) and diffraction losses shown in figure (4). As appearing, the diffraction losses decreasing with increasing of (F) for two lasers because the (F) is depending on resonator length.
Returning to the Eq. (4),the value of ( * ) approach to (0) at a large resonator lengths, so the diffraction losses will be maximum but latter will be minimum when the value of ( * ) approach to (1) because of little resonator lengths. This is explained in Figure (

Conclusions:
It is clear that, the He-Ne laser is more stable than CO2 that is caused by the photons of laser action. In He-Ne laser the photons stay in the optical resonator oscillating between the two mirrors for a long time in comparing with that in CO2 laser because of the high energy of these photons of CO2 laser.