Discussion

the HR surface included AP ring, AP doughnut, AP/RP mode combination and RP ring observed at the output in the order mentioned. The remarkable fact was the appearance of mode combinations with the mixed AP and RP polarizations.

The appearance of the AP ring in the combination with the RP doughnut is also seen in figure 6.2b. To the best of our knowledge, simultaneous generation of modes with the orthogonal axial symmetric polarizations was observed for the first time. Figure 6.4b shows near-field images and corresponding far field intensity profiles of modes in the vicinity of the RP doughnut position (l = 9.04 mm). As the lens is shifted away from the HR surface, RP ring, RP rings, second-order RP mode and RP doughnut are generated in the order listed. Though RP rings and second-order RP mode possessed similar CCD images in near-field, their far-field profiles confirmed the multimode character in RP rings and single-mode nature of the latter.

A sequence of multi-ring beam images (number of rings up to 10) with a central hole was observed in the near and far field for shifts of f = 3.5 cm lens towards HR.

Identical near and far fields of images allow us to consider these beams as resonator modes. Figure 6.4c illustrates far field patterns of the 6-ring mode registered through the rotating linear polarizer. The location of mode lobes relative to arrows (transmission axis of the polarizer) in frames confirms the mode to be azimuthally polarized. Hollow multi-ring modes with RP polarization were also observed with f = 3.5 cm lens when set in combination with c-cut α-BBO. The angular divergence of the observed multi-ring modes was about 2-3 mrad. No modes other then AP or RP doughnuts were observed in the OC region.

6.4 Discussion

The explanation of experimental results may be done using a model of the laser with intra-cavity lens. In order to simplify analysis, we will eliminate the thickness of the Yb-doped plate, ignore aberrations of the intra-cavity lens, the influence of

6.4. DISCUSSION 62 thermal lens in the active medium and initially ‘remove’ the birefringent crystal as well from the cavity. With these assumptions, results of the thin lens res-onator analysis in section 3.4 become applicable to our case. Figure 6.5 shows the schematic of such a plane-plane laser resonator with an intra-cavity thin lens.

Within the framework of geometrical optics, ray trajectories ‘type 1 and ‘type 2 can provide feedback in such a cavity for two different positions of the lens relative to HR mirror. One corresponds to “focusing” (F) and the other to “imaging” (I) condition in the cavity. As per our ray matrix analysis earlier in section 3.4 shows that focusing and imaging configurations are the boundaries of the resonator sta-bility region. Generation on resonator modes occurs for lens positions just between these boundaries. The width of the stability region, (I-F) depends on the resonator length, L and the focal length of the intra-cavity lens, f. The dependence of this width on L and f is shown in figure 6.7 for corresponding experimental values.

Applying this model to our scheme, we find obvious relations of the experimental HR and OC regions of lens shifts to ‘focusing’ and ‘imaging’ configurations and the width W of the separation zone to the width of the laser stability region. Accord-ingly, ray trajectories type 2 and type 1, figure 6.5, correspond to the appearance of “parallel” and “conical” beams at the OC in experiment.

F I

HR OC

Region of stability Type-1 Type-1

Type-2 Type-2

Figure 6.5: Schematic of resonator with intra-cavity lens showing ‘imaging’ (Type-1 trajectory) and ‘focusing’ (Type-2 trajectory) positions of the lens.

RP and AP doughnut mode selection in our laser with an intra-cavity lens

6.4. DISCUSSION 63 and birefringent crystal took place near boundaries of the resonator stability re-gion mentioned above and may be explained as follows. Including the birefringent crystal to the resonator, we get (for most of our experimental conditions) two overlapping stability regions one corresponding to o- and other to e- rays. For resonator with the YVO₄ crystal (refractive indices for o- and e- at λ ≈ 1μm are n_o = 1.96 and n_e = 2.165, respectively [83]) at ‘focusing’ configuration, fig-ure 6.6a, the e-ray will be largely refracted at the crystal surfaces compared to its o-ray counterpart. The distance to the axial focus of the lens for the e- ray (f_e)_ax becomes longer than such a distance for the o- ray, (f_o)_ax < (f_e)_ax. Thus, shifting the lens towards the HR surface of the cavity, the e- ray will reach the

‘focusing’ boundary of its stability region earlier than the o-ray. This means that the e-ray becomes unstable at d = (f_e)_ax. But the o-ray remains in the stabil-ity region and can oscillate. This creates conditions for the selection of only one o-type mode. Because the o-ray corresponds to the azimuthally polarized light, the lowest order azimuthally polarized mode (i.e. AP doughnut) is generated.

Figure 6.6b illustrates selection of the RP polarized doughnut mode in the same cavity with YVO₄ at the ‘imaging’ configuration. Shifting the lens towards OC, the ‘imaging’ boundary of the stability region is initially achieved for the o-rays, which have a shorter distance for ‘imaging’ of the OC onto the HR. The position of the OC image for the e-rays is shown behind the HR surface. This means that o-rays become unstable in the cavity and only e-rays can oscillate. As the e-rays correspond to the radial polarization, the RP doughnut mode is generated from such a cavity configuration. Schemes of mode selection, figure 6.6 are well con-firmed by experimental observations of AP doughnut mode selection at ‘focusing’

and RP doughnut mode at ‘imaging’ configurations in cavities with YVO₄ and lenses of different foci. The observed widths of HR and OC regions, w ≤ 1mm that correspond to oscillations in AP and RP doughnuts, are correlated to the birefringent shift of lens foci, D = (f_e)_ax−(f_o)_ax ≈1 mm for 1 cm³ YVO₄ plate.

Hence, W and w data were used to find the width of the experimental stability

6.4. DISCUSSION 64

e o

c-cut YVO

d = f

f

Pump

e o d = i

i

Pump

HR

HR

OC

OC (a)

(b)

Lens

c-cut YVO

Lens

Figure 6.6: Schemes of mode selection in the laser with intra-cavity c-cut YVO₄ crystal and lens: (a) AP mode at ‘focusing’ (d= (f_e)_ax) and (b) RP mode at ‘imaging’ (d=i_o) lens positions

region. Figure 6.7 illustrates the agreement in the experimental and calculated widths of resonator stability regions (neglecting the small differences to (I-F) aris-ing from birefraris-ingence). It is seen from this figure that for long cavities and lenses of short foci (f ≤ 3.5 cm) the width of the stability region turns comparable or even smaller than the width of the birefringent shift, D. This means that stability regions for o- and e- rays cease to overlap and appear to be separated. In case of (I-F) < D between HR and OC regions no oscillations should be expected. In case ofα-BBO crystal with negative birefringence (refraction indices n_e= 1.58462, n_o = 1.65790 in 1μm region [87]) the mode selective mechanism considered above must be obviously reversed. RP doughnut mode selection should take place at

‘focusing’ configuration and AP doughnut selection- at ‘imaging’ configuration, as observed in experiments.

The observed mode structures, other than AP and RP doughnuts, can be explained by considering conditions of modes competition and modes selection

6.4. DISCUSSION 65

Figure 6.7: Dependence of width of the resonator stability region, (I-F) on the cavity length, L for intra-cavity lenses with different foci (dotted curves, calculations; rhombs-experimental data).

at different lens positions in the cavity. In case of overlapping stability regions for o- and e- rays resonator could support not only modes with AP and RP but

“traditional” scalar modes with linear polarization as well. Conditions for mode selection in this case appeared to be deteriorated. This approach explains features of mode compositions observed in the figure 6.3. At the point of polarization transfer a balance in loss and gain between rays propagated along o-ray and e-ray trajectories could provide the simultaneous appearance of two linearly polarized Laguerre-Gauss modes of LG₀₁ type with the same direction of polarization. It can be assumed that one pair of LG₀₁ lobes was produced by o- ray oscillations and another by e- rays. These four lobes formed a linearly polarized doughnut observed in the experiment in a coexistence with a linearly polarized Gauss-like mode, figure 6.3b. For small lens shifts from the point of polarization transfer, AP or RP doughnut (together with a Gauss mode) oscillated, figure 6.3(a,c).

6.5. CONCLUSION 66