Generation of radially and azimuthally polarized beams in Yb:YAG laser with intra-cavity lens and birefringent crystal

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1 Generation of radially and azimuthally polarized beams in Yb:YAG laser with intra-cavity lens and birefringent crystal Manasadevi P. Thirugnanasambandam, 1,* Yuri Senatsky, 2 and Ken-ichi Ueda 1 1 Institute for Laser Science, University of Electro-Communications, Chofugaoka, Chofu, Tokyo, , Japan 2 P. N. Lebedev Physical Institute of the Russian Academy of Sciences, 53, Leninskiy Prospekt, Moscow, , Russia *manasa@ils.uec.ac.jp Abstract: We demonstrated the operation of cw diode-pumped Yb:YAG laser in radial or azimuthal polarized (RP or AP) beams using a combination of birefringent uniaxial crystal (c-cut YVO 4 or α-bbo) and lens as intracavity elements. RP and AP doughnut modes (M 2 = 2-2.5, polarization extinction ratio :1) with output power up to 60mW were generated. Apart from doughnut modes, RP or AP ring-like off-axis oscillations and multi-ring beams with mixed RP and AP were also observed at the output of this laser scheme. Using intra-cavity short focus lenses with spherical aberrations AP or RP modes of higher orders was obtained. Mechanism of mode selection in the laser is discussed. The large variety of beams with axially symmetric polarizations from the output of the proposed laser scheme may find applications in different fields Optical Society of America OCIS codes: ( ) Optical resonators; ( ) Lasers, diode-pumped; ( ) Modes; ( ) Polarization; ( ) Laser beam shaping. References and links 1. Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon. 1(1), 1 57 (2009). 2. Q. Zhan, Trapping metallic Rayleigh particles with radial polarization, Opt. Express 12(15), (2004). 3. Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, Creation of a three-dimensional optical chain for controllable particle delivery, Opt. Lett. 30(8), (2005). 4. F. Peng, B. Yao, S. Yan, W. Zhao, and M. Lei, Trapping of low-refractive-index particles with azimuthally polarized beam, J. Opt. Soc. Am. B 26(12), (2009). 5. V. G. Niziev, and A. V. Nesterov, Influence of beam polarization on laser cutting efficiency, J. Phys. D 32(13), (1999). 6. A. V. Nesterov, and V. G. Niziev, Laser beams with axially symmetric polarization, J. Phys. D 33(15), (2000). 7. M. Meier, V. Romano, and T. Feurer, Material processing with pulsed radially and azimuthally polarized laser radiation, Appl. Phys., A Mater. Sci. Process. 86(3), (2007). 8. Q. Zhan, and J. R. Leger, Microellipsometer with radial symmetry, Appl. Opt. 41(22), (2002). 9. S. C. Tidwell, G. H. Kim, and W. D. Kimura, Efficient radially polarized laser beam generation with a double interferometer, Appl. Opt. 32(27), (1993). 10. M. Stalder, and M. Schadt, Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters, Opt. Lett. 21(23), (1996). 11. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, Tailoring of arbitrary optical vector beams, N. J. Phys. 9(3), (2007). 12. V. G. Niziev, R. S. Chang, and A. V. Nesterov, Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer, Appl. Opt. 45(33), (2006). 13. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, The formation of laser beams with pure azimuthal or radial polarization, Appl. Phys. Lett. 77(21), (2000). 14. T. Moser, M. A. Ahmed, F. Pigeon, O. Parriaux, E. Wyss, and T. Graf, Generation of radially polarized beams in Nd:YAG lasers with polarization selective mirrors, Laser Phys. Lett. 1(5), (2004). 15. M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers, Opt. Lett. 32(22), (2007). (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1905

2 16. I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, 2 kw, M 2 < 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers, Opt. Lett. 32(1), (2007). 17. D. Pohl, Operation of a ruby laser in purely transverse electric mode TE 01, Appl. Phys. Lett. 20(7), (1972). 18. J.-F. Bisson, J. Li, K. Ueda, and Yu. Senatsky, Radially polarized ring and arc beams of a neodymium laser with an intra-cavity axicon, Opt. Express 14(8), (2006). 19. K. Yonezawa, Y. Kozawa, and S. Sato, Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal, Opt. Lett. 31(14), (2006). 20. K. Yonezawa, Y. Kozawa, and S. Sato, Compact Laser with Radial Polarization Using Birefringent Laser Medium, Jpn. J. Appl. Phys. 46(No. 8A), (2007). 21. R. Zhou, B. Ibarra-Escamilla, J. W. Haus, P. E. Powers, and Q. Zhan, Fiber laser generating switchable radially and azimuthally polarized beams with 140 mw output power at 1.6µm wavelength, Appl. Phys. Lett. 95(19), (2009). 22. J.-L. Li, K. Ueda, L.-X. Zhong, M. Musha, A. Shirakawa, and T. Sato, Efficient excitations of radially and azimuthally polarized Nd 3+ :YAG ceramic microchip laser by use of subwavelength multilayer concentric gratings composed of Nb 2O 5/SiO 2., Opt. Express 16(14), (2008). 23. W. Koechner, Solid-State Laser Engineering (Springer science + business media, Inc., sixth revised and updated edition, 2006), Chap N. Hodgson, and H. Weber, in Laser Resonators and Beam Propagation, (Springer science + business media, Inc., second edition, 2005). 25. M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser, Laser Phys. Lett. 7(9), (2010). 26. F. D. Vanderwerf, Applied Prismatic and Reflective Optics, (SPIE Press, 2010), Chap Yu. Senatsky, J.-F. Bisson, A. Shelobolin, A. Shirakawa, and K. Ueda, Circular modes selection in Yb:YAG laser using an intracavity lens with spherical aberration, Laser Phys. 19(5), (2009). 28. Y. Kozawa, and S. Sato, Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler, Opt. Lett. 33(19), (2008). 29. Y. Kozawa, and S. Sato, Sharper focal spot formed by higher-order radially polarized laser beams, J. Opt. Soc. Am. A 24(6), (2007). 1. Introduction During the last decade there has been an increasing interest in laser beams with cylindrical symmetry in polarization. Properties of these so-called cylindrical vector (CV) beams have been the topic of numerous studies [1 16]. The most famous representatives of CV beams - doughnut modes with radial and azimuthal polarizations (RP, AP) have proved to be potential for applications in various research fields including optical trapping and micromanipulation [2 4], material processing [5,7], high-resolution metrology [8], particle acceleration [9]. Numerous methods have been proposed to generate such RP and AP doughnut beams either passively (out of laser cavity) [9 12] or actively (in the laser cavity) [13 16]. Solid-state lasers with grating mirrors or thermally induced birefringence in resonators have been shown to generate AP and RP beams with output powers ranging W for applications in laser technology [15,16]. The first experimental study on beams with axially symmetric polarization was carried out in a pulsed ruby laser, where the discrimination of AP doughnut near the stability limit of the resonator was made possible by using an intra-cavity c-cut calcite crystal having an axially symmetric birefringence [17]. Recently, birefringent crystals (undoped and Nd 3+ - doped c-cut YVO 4, GdVO 4 or calcite) for AP/RP mode selection has taken a comeback in several works with Nd- solid-state and Er- fiber lasers operating in a cw regime at a moderate power of mW, which is of interest for many applications [18 21]. These schemes also exploited double refraction to discriminate one of the two orthogonal polarizations by inducing higher losses to either the extraordinary (e) or ordinary (o) ray which propagate along slightly different off-axis trajectories in the laser cavity. In the present work, we propose a simple diode-pumped Yb:YAG laser scheme with resonator enclosing the lens and crystal with either positive (YVO 4 ) or negative (α-bbo) birefringence that permits to obtain at λ 1030nm wavelength, doughnuts of both AP and RP with a good beam quality and polarization contrast. Switching between RP and AP doughnut outputs can be achieved by a horizontal translation of the intra-cavity lens along the resonator axis. Features of mode compositions in the region of polarization transformation are discussed. Higher order CV modes, RP or AP ring-like off-axis oscillations, and multi-ring (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1906

3 beams with mixed RP and AP polarizations were also observed at the output of this laser scheme. Mechanism of mode selection in a lens resonator with birefringent uniaxial crystal is discussed. 2. Experimental setup Fig. 1. Schematic of the laser set up. Figure 1 shows the schematic of a laser resonator (length, L = cm) formed by the highreflection (HR) surface (for λ 1030nm) of the plane-parallel Yb:YAG (9.8 at.% doped) ceramic plate (9x11x1.5)mm 3 and the plane output coupler (OC) of 98% reflectivity. The other surface of the Yb:YAG plate was anti-reflection (AR) coated for λ 1030nm. The plate was conductively cooled between copper slabs and end-pumped by a cw fiber-coupled 940nm laser diode at room temperature. 1-4W pump radiation was focused in the plate to 100μm diameter spot. Optically uniaxial crystal (c-cut YVO 4 or α-bbo) (10x10x10)mm 3 with AR coated plane-parallel faces was placed at a distance 1-4cm from HR surface such that its optic axis was aligned along the resonator axis. Plano-convex glass lenses of 25mm diameter and focal lengths, f = 3.5; 5; 7.5; 10 or 20cm were used through the experiments. The lens initially was placed at the position d f ax (d - distance between the lens apex and HR surface, f ax -the distance to the focus of paraxial rays at HR, which was determined for every combination of lenses and crystals directly in the cavity using a reference linearly polarized He-Ne laser beam). The lens could be moved along the resonator axis on a horizontal translation stage. The fine positioning of the lens was done in steps of 10μm using an additional micrometer stage. In some experiments, a circular diaphragm (diameter 3 to 5mm) was placed between the lens and OC. The Yb:YAG laser operated at 1030nm in cw regime. The laser output could be monitored using a power meter and a charge-coupled device (CCD) camera placed either beyond the OC (to record the near-field pattern) or at the focus of an extra-cavity lens (to record the far-field). Near field images of modes (with reduction 0.5x) were also recorded by placing the camera in front of the focus of an extra-cavity lens. The size of the CCD camera frame is mm 2. Over 80% of pump radiation was absorbed by the Yb:YAG plate. In order to prevent the residual pump radiation beyond the OC from reaching the power meter or CCD camera, external selective filters were used. 3. Experimental results The cylindrical symmetry obtained on precision alignment of the cavity provided conditions for the generation of various circular modes by shifting the lens about its initial position, d f ax along the resonator axis. There were found two regions where AP and RP doughnut modes appeared: one of these regions was closer to HR (at lens positions d < f ax ), the other was shifted further towards OC (at lens positions d > f ax ). These regions will be referred conventionally as the HR region and OC region respectively. In these regions, the doughnuts remained stable over small lens shifts, l 1mm. For lens positions between these two regions intensity profiles with a central hole (that is a distinguishing characteristic of a doughnut) were distorted. Figures 2a 2d shows four groups of CCD camera images of AP and RP doughnut modes in near-field which were obtained using c-cut YVO 4 and α-bbo crystals (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1907

4 in the resonator of the length, L = 114cm with the lens f = 10cm sans any intra-cavity diaphragm at pump power about 1W. Position of the lens relative to the HR surface in each case, d, is indicated alongside the corresponding sequence of images. Each group of images included frames; one registered directly at the output and the others recorded through a linear polarizer (arrows indicating the transmission axis in each case) placed in front of the camera on a rotating mount. The orientation of mode lobes relative to the arrows in the frames clearly classifies them as AP (a, d) and RP (b, c) doughnuts. Intensity profiles of doughnuts in far field were similar to their near field patterns. In some cases, doughnuts were observed with an admixture of modes of higher-order, Fig. 2c. The intensity of these modes increased with pump power. But these higher order modes could be easily removed using an intra-cavity diaphragm. With intra-cavity c-cut YVO 4, AP doughnut mode was observed in the HR region, Fig. 2a. The RP doughnut mode oscillated in the OC region, Fig. 2b. Contrastingly, resonator with intra-cavity c-cut α-bbo generated RP doughnut mode in the HR region and the AP mode in the OC region, Figs. 2c, 2d. Near the threshold of generation, at 1W pump power, the Yb:YAG laser output power (doughnut mode) was 6mW. Using an intra-cavity diaphragm it was possible to produce doughnut beams of high quality with output power over 60mW at 4W pump power. Fig. 2. CCD camera images in near-field of AP and RP doughnut modes generated from Yb:YAG laser cavity with birefringent (a,b) c-cut YVO 4 or (c,d) α-bbo crystals and lens f = 10cm placed at distance d<f ax or d> f ax from HR surface. The transmission axis of linear polarizer placed before the camera is indicated by arrows in the corresponding frames wherever used. The width of the HR or OC regions that allowed oscillations in doughnut modes was w 0.8mm while the width of the entire zone separating the HR and OC regions was W 11mm. Similar AP/RP doughnut modes were observed using other lenses. While the width of the HR and OC regions appeared to match in all cases with w 1mm, the separation zone between these regions, W, showed dependence on the focal length of the lens used. For example, in cavity L = 114 cm, W 1mm was observed with lens f = 5cm, whereas W 6mm was observed with f = 7.5 cm. The width, W was also affected by the cavity length L. For the cavity with lens f = 5cm, the separation zone widened, W 3mm when length of cavity was reduced to L = 71 cm. Very narrow W in the range mm were observed in cavities of L = cm with f = 3.5cm lens. For lens placed within the HR region, the laser generated doughnut modes with spot diameters 2-4mm at the output. Angular divergence of these modes was mrad. However, in the OC region, spot sizes of beams generated measured 1mm or smaller and had (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1908

5 a larger divergence up to 5mrad. We indicate beams with smaller and larger divergences conventionally as parallel and conical beams. So, parallel beams were confined within the HR region and conical beams existed in the OC region. Spot sizes and divergence of conical beams could be changed within a small range of lens shifts within the OC region. It was also observed that the divergence of conical beams could be compensated to 1mrad using an extra-cavity lens. Propagation factor for doughnut modes, M 2 was determined based on measurements of the Rayleigh range of the focused beam [22] yielding M 2 = (theoretical M 2 value for doughnut modes, M 2 = 2 [23]). The polarization purity of output doughnuts was measured (both for parallel and conical beams) according to [18] using a 300-μm-wide slit and the polarization analyzer mounted on rotation stages. Polarization Extinction Ratio (PER) values in the range of 50-90:1 was registered. The use of a circular 3-4mm diaphragm in the cavity helped to improve polarization contrast. The highest PER values, up to PER = 100:1 were observed for conical beams. Modes of other types were also found inside and outside of 2 regions of AP and RP doughnuts appearance. In the separation zone when the width was as wide as W 10mm (for f = 10 and 20cm) along with lens shifts, the axial minimum of the mode intensity gradually transformed to bell-shaped beam profiles at the output. Such profiles (unpolarized or with a linear polarization) dominated almost over all the length of the separation zone. For W < 5mm, it was possible to follow the transformation of oscillations from one polarization state to the orthogonal one. Figure 3 shows recordings of one such transformation from AP to RP observed at the output from cavity, L = 114cm, enclosing lens f = 5cm, c-cut YVO 4 crystal and intra-cavity diaphragm of 3mm diameter for lens shifts between the HR and OC regions. Displacements of the lens are indicated by micrometer readings on the lens translation stage, l. Using the extra-cavity polarizer it was possible to observe (shifting the lens in little steps) the transition from the mode with the dominant AP-like, Fig. 3a to the RP-like mode, Fig. 3c. Inside the separation zone, there was a definite position of lens indicative of AP to RP transfer, Fig. 3b, where the mixture of doughnut and Gauss-like modes linearly polarized in orthogonal directions was observed. Doughnuts or mode compositions with two off-axial minima were observed at polarization transfer in cavities with lens of f = 3.5cm. Fig. 3. Mode profiles and polarization change observed on shifting the lens f = 5cm in the Yb:YAG laser cavity between HR and OC regions. Apart from the doughnut beam profiles, several mode structures were observed in the HR region by shifting the lens towards HR or OC in cavities without a diaphragm. Figure 4 shows CCD camera images of near and far field patterns of output beams in the resonator L = 114cm: with c-cut YVO 4 and the lens f = 10 cm (a), α-bbo and the lens f = 7.5 cm (b), and with the lens of f = 3.5 cm and the YVO 4 (c). Images were recorded using extra-cavity lens of f = 1m. Micrometer readings, l for schemes with different intra-cavity elements are not correlated with each other; however they indicate the displacement of lens between each output profile. Figure 4a shows CCD images in near-field of modes along with their corresponding images captured at the CCD with a linear polarizer (transmission axis indicated by arrows) in the vicinity of the AP doughnut position (l = 9.24mm). The series of modes (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1909

6 generated in this case by lens shifts away from 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 Fig. 2c. To the best of our knowledge, observation of mixed radial and azimuthal polarizations in a single output has been reported for the first time. RP ring- and arc-like off-axis oscillations were obtained previously in Nd:YAG laser with an intra-cavity axicon [18]. Figure 4b shows near-field images and corresponding far filed intensity profiles of modes in the vicinity of the RP doughnut position (l = 9.04mm). 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. Fig. 4. Sequence of modes registered at the output of laser (L = 114cm) by shifting intra-cavity lens: (a) near field images with YVO 4 and lens f = 10cm; (b) near field images and far field intensity profiles with α-bbo and lens f = 7.5cm; (c) far field images with YVO 4 and lens f = 3.5cm. 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 = 3.5cm lens towards HR. Identical near and far fields of images allow us to consider these beams as resonator modes. Figure 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 = 3.5cm lens when set in combination with c-cut α-bbo. The angular divergence of the observed multi-ring modes was about 2-3mrad. No modes other then AP or RP doughnuts were observed in the OC region. (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1910

7 4. Discussion The explanation of experimental results may be done using a model of the laser with intracavity lens. A number of studies are devoted to lasers with intra-cavity lens [23,24]. 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 thermal lens in the active medium and initially remove the intra-cavity crystal as well from the cavity. With these assumptions, results of the thin lens resonator analysis [23,24] become applicable to our case. Figure 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, Fig. 5. A ray matrix analysis shows that focusing and imaging configurations are the boundaries of the resonator stability region. Generation on resonator modes occurs for lens positions just between these boundaries [23 25]. 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 Fig. 6 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. Accordingly, ray trajectories type 2 and type 1, Fig. 5, correspond to the appearance of parallel and conical beams at the OC in experiment. Fig. 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 and uniaxial crystal took place near boundaries of the resonator stability region 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 YVO 4 crystal (refractive indices for o- and e- at λ 1μm are n o = 1.96 and n ex = 2.165, respectively [19]) at focusing configuration, Fig. 7a, 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 stability 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 7b illustrates selection of the RP polarized doughnut mode in the same cavity with YVO 4 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, Figs. 7a, 7b are well (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1911

8 confirmed by experimental observations of AP doughnut mode selection at focusing and RP doughnut mode at imaging configurations in cavities with YVO 4 and lenses of different foci. Fig. 6. 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). 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 1mm for 1cm YVO 4 plate according to calculations in [20]. Hence, W and w data were used to find the width of the experimental stability region. Figure 6 illustrates the agreement in the experimental and calculated widths of resonator stability regions (neglecting the small differences to (I-F) arising from birefringence). It is seen from this figure that for long cavities and lenses of short foci (f 3.5cm) 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 of α-bbo crystal with negative birefringence (refraction indices n e = , n o = in 1μm region [26]) 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. (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1912

9 Fig. 7. Schemes of mode selection in the laser with intra-cavity c-cut YVO 4 crystal and lens: (a) AP mode at focusing (d = (f e) ax) and (b) RP mode at imaging (d = i o) lens positions. In the above mode selection scheme, Fig. 7, the intra-cavity lens was assumed ideal or free from spherical aberrations; when in fact, we used plano-convex lenses with spherical aberrations. In addition to AP and RP mode discrimination at boundaries of the resonator stability region, spherical aberrations of intra-cavity lenses also play a role in mode selection mechanism in our laser [25] by suppressing of fundamental Gaussian mode and selection in AP and RP doughnuts. In the overlapping stability regions for o- and e- rays, the resonator can support not only modes with AP and RP but also traditional scalar modes with linear polarization as well. In between focusing and imaging configurations, the feedback in paraxial rays serves oscillations in linearly polarized fundamental Gaussian mode, Fig. 3. For appreciable lens shifts in the HR and OC regions, towards and away from the HR surface respectively, the feedback in paraxial rays and therefore the generation in a Gaussian mode are terminated. Due to lens aberrations instead of paraxial feedback, a feedback for rays refracted from the peripheral annular zones on the lens spherical surface appears. Since such lens shifts allow stable oscillations along these annular regions to be consistently focused at the mirror surface, providing a peripheral feedback, generation in ring-like modes (e.g. doughnuts) appears. The observed mode structures, other than AP and RP doughnuts, can be explained by considering conditions of modes competition and modes selection at different lens positions in the cavity. This approach explains features of mode compositions observed at Fig. 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 LG 01 type with the same direction of polarization. It can be assumed that one pair of LG 01 lobes was produced by o- ray oscillations and another by e- rays. These four lobes formed a linearly polarized doughnut and it appeared in coexistence with a linearly polarized Gauss-like mode, Fig. 3b. For small lens shifts from the point of polarization transfer, AP or RP doughnut (together with a Gauss mode) oscillated, Fig. 3a, 3c. AP and RP ring-like beams and combinations of rings with doughnuts observed at focusing conditions in cavities with YVO 4 or α-bbo (no intra-cavity diaphragms) are just examples of typical multi-mode oscillations, similar to oscillations observed in resonators with aberrated intra-cavity lenses and axicons [18,25,27]. In the resonator with a uniaxial crystal, near the stability limit, the presence of aberration provides conditions for the appearance of ring-like oscillations and mode compositions with mixed AP and RP polarizations, Fig. 4. The formation of ring- like structures in a lens resonator was determined by the feedback in parallel beams (type 2, Fig. 5). No ring-like oscillations were observed due to the feedback of type 1 (in conical beams). The absence of competing modes in (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1913

10 different polarizations at the imaging configuration explains the larger polarization extinction ratio, PER observed for conical AP and RP doughnut beams. When spherical aberrations are more pronounced as in case of f = 3.5cm, AP and RP multi-ring modes appear. The mechanism of higher order Laguerre-Gauss mode selection in the Yb:YAG laser with an intra-cavity f = 2.5cm lens with spherical aberration was described in detail [25]. Similar mechanism should work in our cavity with the uniaxial crystals (YVO 4 or α-bbo) and f = 3.5cm lens. Shifts of the f = 3.5cm lens towards HR provide (due to aberrations) the feedback for AP or RP radiation going through annular ring zones with successively increasing diameters on the lens surface. Selection of higher order AP or RP modes becomes feasible in these conditions. To the best of our knowledge generation of AP modes of higher orders, Fig. 4c is reported for the first time. Observation of RP modes of the 2nd and 3rd order was reported in [28]. Following the classification of CV modes introduced in [6,29], the multi-ring modes observed in our experiments may be classified as A-TEM* p1 and R-TEM* pl type modes. 4. Conclusion In conclusion, a simple diode-pumped cw Yb:YAG laser scheme with an intra-cavity lens and either positive or negative birefringent uniaxial crystal has been successfully demonstrated to produce RP or AP doughnut modes with output power up to 60mW. The output can be easily switched between the radial and azimuthal polarization by horizontal translation of the intracavity lens. PER of the output up to 100:1 was observed. Propagation factor, M 2 = confirms mode purity. Apart from the fundamental RP and AP doughnut modes, various other circular mode structures including AP or RP rings and modes combinations with mixed polarizations (AP and RP) can be produced from the same scheme. Using intra-cavity short focus lenses with spherical aberrations together with the uniaxial crystal a sequence of hollow AP or RP modes of high orders have been obtained. This scheme for RP and AP modes generation can be improved in efficiency and also may be transformed for use with lasers of other types in cw and pulsed regime. The large variety of beams with axially symmetric polarizations from the output of the proposed laser scheme may find interesting applications in different fields. Acknowledgements The authors thank J. Li for helpful discussions. M.P.T acknowledges the support of Japanese Government (Monbukagakusho) scholarship. This work was supported partly by the 21st Century COE Program on Coherent Optical Science and Asian Core Program of the Japanese Society for the Promotion of Science. (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 1914