of peaked and hollow multi-ring beams at the distance S≈10 m from the laser resonator. The half-widths of central hole or peak in all beams at ≈ 10 m were 1- 3 mm. The angles of divergence of these beam diameters were estimated 2-5×10−3 rad. But the estimated angular divergence of the central parts of these beams estimated from their diameters at the output coupler and at 10 m from the resonator was 0.9-3×10−4 rad; an order of magnitude lesser than the diffraction limit associated with the half-widths of these central parts at near-field.In other words we observed NDF propagation of central parts of these beams. Such NDF propagation of central parts beams was observed over 40 m distance. In accordance with measurements carried out along the beam propagation distance, when the full beam power of 4 mW, the power concentrated in the central peak of the multi-ring beam was constant throughout the propagation distance, about 0.2 mW.
Apart from the multi-ring beams, NDF properties were also found associated with doughnut-like mode, figure 5.3c. The divergence of the central minimum or hole is ten times smaller than the divergence of the doughnut as a whole. When the cavity offered selection in pure LG modes even in the absence of obstacles, they transformed into multi-ring beams with NDF characteristics by a small increase in the pump power, which shows their mode stability.
5.3 Discussion
An analysis of our experimental data on LGpl mode selection can be done within the framework of a simplified model. We will eliminate the thickness of the active medium in our resonator and consider the active medium located directly at HR mirror for the time being. Under this assumption, we will also ignore the possible influence of the thermal lens in the active medium on the resonator parameters.
If the optical power of a thermal lens is small relative to the optical power of the intra-cavity elements, the laser beam must not be strongly affected by the thermal lens.The estimation of the thermal lens focal length, fth, may be done
5.3. DISCUSSION 49 using the data reported earlier [81]. In this study the thermal lens, fth = 12cm was measured for a 1.95mm long, 8% doped Yb:YAG sample at 10W pump power.
In our experiment using similar Yb:YAG sample, at only 1-2W pump power with a strongly aberrated lens f = 2.5 cm, we must have a relation f<< fth. So we may ignore the influence of the thermal lens on the resonator and laser beam parameters.
The laser resonator theory does not consider the case of an intra-cavity thick lens with strong spherical aberration (Chapter 3). The stability conditions of the resonator with a thin lens can be employed for the case of an intra-cavity aberrating lens, which may be considered as a sequence of thin annular lenses. It follows from calculations in chapter 4 that for every position of the thick aberrating lens in the resonator, a particular thin annular zone at this lens surface corresponds to the resonator stability region and provides favorable conditions for the mode selection. Shifting the lens towards the HR mirror, the operating annular zone characterized by the refracting angle γ and radius r shifted to the lens periphery (γ and r increased). At the same time the width of the oscillating mode in the active medium near the HR mirror decreased. In order to fit this new transformed geometry of the cavity, a transfer occurred from the previous oscillating LGpl mode towards a new mode with increased p and l indices. According to this mode selection mechanism the resonator with the movable intra-cavity aberrating lens supported generation of LGplmodes of all observed orders in a single experimental set-up. However, further shifts of intra-cavity lens without intra-cavity obstacle allowed formation of multi-ring modes very different from the profiles of classical LGpl modes exhibiting an order of magnitude smaller divergence associated with central parts of the modes. Also, NDF nature was associated with doughnut-like mode; whose intensity profile is very different from that of the LG∗10 mode.
5.3. DISCUSSION 50
5.3.1 Coupling of cavity modes for NDF propagation
To establish a more clear explanation to the observation of NDF characteristics associated with the central parts of certain mode profiles, we refer back to the ear-lier work by J-F.Bisson et al. [82], figure 5.1 where the intra-cavity axicon placed with its line focus at the Nd-dope gain element generated ring and arc beams at the output. Our current model for a resonator with intra-cavity lens is very similar to the former case with oscillations in the cavity occurring in two distinct patterns
‘type-1’ and ‘type-2’. But the output from such a cavity consisted of incoherent oscillations along the two feedback loops. It was possible to cut oscillations along
’type-1’ using a diaphragm. To check the role played by the two feedback loops in the intra-cavity lens resonator, an experiment was performed using intra-cavity obstacles (metal spheres mounted on thin glass plates) of different sizes (1-2 mm diameter) inside the resonator. When the resonator was set to oscillate on its own without any intra-cavity obstacles. Multi-ring mode exhibiting NDF char-acteristics was observed at the output, figure 5.4a. When a spherical obstacle of 1.7 mm diameter was introduced along the resonator axis between the lens and the output coupler, to obstruct paraxial oscillations; the laser offered selection in pure Laguerre-Gaussian mode (figure 5.4b) as observed in chapter 4. When the diameter of the obstacle was 1.2 mm, CCD image recorded a mode profile which held resemblance to interference of figure 5.4a and b. Throughout the experiment all other experimental conditions were maintained constant. This experimental evidence establishes a coupled mode performance in the cavity when there were no obstacles. The reason for the presence of coupling between the cavity modes in the current scheme and its absence in the intra-cavity axicon laser scheme comes from the differences in the gain element used. Aperture guiding mechanism that is inherent of end-pumped quasi-three level laser schemes checks on the mode ra-dius at the HR surface trying to keep it as small as possible and reduce diffraction losses, thus enhancing mode coupling. The cavity modes behave more or less like a
5.3. DISCUSSION 51
(a) (b) (c)
Figure 5.4: CCD camera images captured in near-field (a) with 1.7 mm obstacle (b) 1.2 mm obstacle (c) no obstacle along the resonator axis.
master-slave oscillator with feedback along the ‘focusing condition’ acting like the
‘master’ and along the ‘imaging condition’ acting like a ‘slave’; in which case the oscillations in the central part of the output (type-1 feedback) are coupled with its surrounding rings (type-2 feedback) giving it near-diffraction-free properties.
The possibility of mode coupling in cavity also arises in the cavity because of the mode selective element used - the aberrated lens. Estimations based on geometrical optics were done for cavity length, L = 114 cm and fax= 25 cm. When the shift of lens was large enough to allow the ‘stability region’ on lens to shift to y co-ordinate of lens from resonator axis (y) = 4 mm, the angles subtended by the stable ‘parallel’ and ‘conical’ rays γp and γc at the HR surface of Yb:YAG are estimated. The ray tracing schematic is shown in figure 5.5. The small difference Δγ = γc −γp and taking into account that the rays in fact do not have strict boundaries as they propagate inside the resonator, it is clear that mode coupling is possible in the cavity. In the presence of an aberrated thermal lens (which exists in an end-pumped system), Δγ becomes even smaller as the peripheral rays are bent further to coincide at the HR surface as compared to its paraxial counterpart.
The possibility of coupling at far shifts of lens towards the HR (multi-ring beams), and near fax(doughnut-like mode) and in long cavity lengths is because of the narrow stability region in these cases that results establishing the possibility of mode coupling.
The dominant selection in pure Laguerre-Gaussian modes even without ob-stacles in certain cases is because of the initial conditions of the resonator (lens