1. Introduction

Cylindrical vector beam (CVB), including azimuthally and radially polarized (AP, RP) beams, characterized by cylindrical symmetric spatial polarization distributions across the beam cross-section have attracted considerable attention of many important applications. Such as: particle trapping and acceleration, high-resolution microscopy, material processing [1–6] and so on.

To produce CVBs with AP or RP, many researchers have been developed different extra-[7–10] and intra-cavity [11–14] approaches in the last decades. One simple way consists in placing a birefringent crystal inside the resonator with the conical beam, because this crystal is low-cost and available everywhere, and the spatial walk off along with it provides us a simple and efficient discrimination mechanism for cylindrical vector mode in a wide range of wavelength.

In this letter, we present the intra-cavity generation of beams with both AP and RP in an Nd:YAG laser. A c-cut YVO₄ crystal was employed as intra-cavity polarization discriminator that distinguishes an azimuthally polarized beam from a radially polarized one, or visa versa. The conversion between AP and RP in a laser by only changing the input pump power was studied. The details are shown as follows.

2. Experimental setup

The experimental scheme is shown in Fig. 1. A Φ20 × 1.2 mm, 1% at. Nd:YAG was used as active materials, both surfaces of it were anti-reflection (AR) coated at 808 nm and 1064 nm. This microchip was also sandwiched between two flat copper plates, and both copper plates had the Φ2 mm diameter light tunnel drilled along the cavity axis. The copper plate was connected to the 15 °C water cooling. The resonator was formed by the high-reflection-coated (HR) mirror, two lenses (L₁ and L₂) and the output coupler (OC) mirror. The HR mirror was a plane-concave mirror with 50 mm curvature radius, and its plane surface was anti-reflection coated at 808nm and its concave surface was anti-reflection coated at 808nm and high-reflection coated at 1064nm. The distance between HR and L₁ was 100mm. Nd:YAG was in the middle of them. L₁ and L₂ were two plano-convex lenses with respective focal length of 50mm and 40mm and with both surface of them AR coated at 1064 nm, and they were used to collimate and refocused the round-trip light. The distance between L₁ and L₂ was 40mm. The OC was a plane mirror coated with 95% reflectivity at 1064nm, and it could be moved along the resonator axis on a horizontal translation stage with a minimum step of 10μm. With such setup, there formed two beam waists in the cavity, and first waist was located in between HR mirror and L₁ and identical to the position of the laser crystal, while the second waist was exactly on the reflectance surface of OC. In this way, a conical beam was formed between L₂ and OC in the resonator. The pump light from a fiber (400μm) coupled laser diode (λ = 808 nm) was focused by a coupler (1:1) in the sample along the resonator axis. The converging pump light passed through HR mirror into Nd:YAG and formed a spot of about r_P = 300μm (r_P is the focal spot radial of pump beam in the crystal), and formed the end-pumped structure.

 

Fig. 1 Schematic of experimental setups for the azimuthally or radially polarized Nd:YAG laser.

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An optically uniaxial c-cut YVO₄ crystal was placed behind L₂ as the polarization discriminator in this cavity. It had 8 × 8 mm cross section and 20 mm length with both surface of it AR coated at 1064 nm. The YVO4 crystal has positive birefringence, and the refractive indices for ordinary and extraordinary rays (e-ray and o-ray) are n_o = 1.9573 and n_e = 2.1652 at 1.064 nm, respectively. When the beam passing the birefringent crystal is diverging or converging, the e-ray and o-ray will take different paths because of the different refractive indices. The forward conical beam coming from L₂ side is separated into the e- and o-ray components and then they would converge at different points along the cavity axis. Because the e-ray will be largely refracted at the crystal surfaces compared with the o-ray, as shown in Fig. 1, the limit of the stable cavity length for the e-ray becomes longer than that for the o-ray. Thus, a cavity length range (that is the OC’s position) in which the e-ray is stable but the o-ray is unstable will be generated, or visa versa.

3. Results and discussion

According to Fig. 1, the OC’s position is determined by following experimental steps. At a position, only one of o- and e-ray could be retro-reflected into resonator with minimum round-trip loss, and therefore it is expected that a desired vector mode can be excited. So when the OC was positioned near the theoretical (estimated) converging (focal) point of o-ray, the laser began to oscillate in doughnut-shaped mode within the pumping level above the lasing threshold. Then we gradually moved the OC away from its initial position along the resonator axis. Figure 2 shows far-field intensity distributions when the cavity length was gradually increased which were obtained. As shown, in two regions the annular intensity distributions with central null were clearly discerned. For one of these two regions the OC was closer to YVO₄, for the other the OC was shifted further from YVO₄. Both regions will be referred conventionally as the ‘near region’ and ‘far region’, respectively. In these two regions, the doughnuts remained stable. This result is consistent with the above theoretical analysis. That is to say, as Fig. 1, the doughnut mode obtained in the ‘near region’ is formed by the o-ray, and the doughnut mode obtained in the ‘far region’ is formed by the e-ray.

 

Fig. 2 Variation of intensity distributions of laser beams at different cavity length.

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Further, the polarization states of the doughnut-shaped laser modes in two regions were checked by recording the intensity distribution through a linear polarizer analyzer under different orientations. Figures 3(a) and 3(b) depict the far- and near-field full beam profiles. Figures 3(c)–3(f) show the corresponding far-field intensity distribution of laser beam after the polarizer when the polarizer was rotated at different orientations. As seen, in the “near region” the symmetric two-lobe patterns were always perpendicular to the respective polarizer axis, and this phenomenon manifested that the doughnut-shaped laser beam was AP. In the “far region” the symmetric two-lobe patterns were always parallel to the respective polarizer axis, and this phenomenon manifested that the doughnut-shaped laser beam was RP. It means that by simple adjusting the OC’s position (cavity length) we obtained AP beam at short L and RP beam at long L with YVO₄ in this Nd:YAG laser.

 

Fig. 3 In both near and far region, (a) far- and (b) near-field intensity distribution of the full beam profile; (c–f) Intensity profiles of far-field laser beam transmitted through the polarizer analyzer. The black arrows direction indicate the respective orientations of the polarizer analyzer’s axis.

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When the laser began to oscillate in the “near region”, the cavity was optimized to maximize the output power at the threshold P_abs = 4.29W. As a result, the optimum position of OC was fixed to that corresponding to a cavity length of L = 178.7 mm. The laser power always increased linearly with P_abs above threshold pump power. But we found that the doughnut mode was not always maintained as the pump power increased. Figure 4 depicts the captured total intensity distributions of laser beams at different P_abs. In this case, the series of modes generated by the absorbed pump power P_abs increases from 4.29 W to 6.78W included doughnut mode, admixture of mode and doughnut observed at the output in the order mentioned. As shown, the absorbed pump power P_abs lower than 5.07 W the laser beam was a doughnut AP beam. When P_abs exceeded 5.07 W, the doughnut mode pattern suddenly changed to a multimode. Then P_abs increased continuously to 6.4 W, the laser mode suddenly changed to a doughnut mode again. At this time the second doughnut beam was proved to be RP beam. We are surprised to find that there were also two cavity mode changes (doughnut mode change to another mode and then to doughnut mode again) only with pump power increase from low to high; and the doughnut modes, in the doughnut mode appeared regions, were proved to be AP (at low pump power region) and RP (at high pump power region), respectively. All those doughnut modes remained stable. The experiment results show that in this laser the laser mode can be converted between AP and RP only by changing the input pump power.

 

Fig. 4 Far-field intensity distributions of laser beams at P_abs = 4.29W(a), 4.73W(b), 5.07W(c), 5.54W(d), 6W(e), 6.4W(f), 6.78W(g), respectively, at L = 179.8 mm.

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According to experiment results of Fig. 4, we can make a preliminarily inference that at other suitable cavity lengths there will also be laser mode conversion with the pump power change. As is well known in a laser the conversion between two cavity modes is due to changes in the stability conditions of two modes. In our experiment, there was no other reason for mode conversion except the change of the input pump power. So this phenomenon of lase mode transition was attributed to the variation of cavity configuration with thermal lensing effect [15,16]. More detailed theoretical explanation for the mechanism is given in the following section.

As we know, the temperature gradient in an end-pumped laser crystal due to an inhomogeneous pumping (heating) produces thermal strain and a transverse gradient of refractive indices. For a paraxial coherent beam propagating in the heated laser crystal, thermal lens focus length (f_th) for o-ray (AP beam) and e-ray (RP beam) is derived as [17]

Where A is area of crystal, l is the length of crystal, r₀ is the radial direction of crystal along the cavity axis, P is availability conduce thermal in crystal, P = ηP_abs, η is an efficiency factor which relates the absorbed pump power to the power dissipated as heat in crystal. According to Eq. (1), for Nd:YAG crystal we used the thermal conductivity K = 0.14 W/cm·$℃$, the thermal dispersion $dn/dT=7.3\times {10}^{-6}/°\text{C}$, coefficient of thermal expansion$\alpha =7.5\times {10}^{-6}/°\text{C}$, function of elasto-optical coefficient for radial or azimuthal $C_r=0.017$, $C_a=-0.0025$, and index of refraction n₀ = 1.82 [17]. A ray transfer matrix analysis [18] may be applied for every resonator configuration. Such an analysis should permit finding the region of resonator stability and mode radii at the Nd:YAG for every configuration with the different pump power. In order to simplify the analysis we will initially eliminate the thickness of the Nd:YAG disk and ignore aberrations of the intra-cavity lens. As well known, the fundamental mode has the smallest beam radius in the resonator. The beam radius of each mode increases with increasing mode number. However, the ratio of the radii of the higher order mode to the fundamental mode remains unchanged in any plane inside or outside the cavity, in both near and far fields. In other word, the radii of the higher order mode and fundamental mode have the same features. So with the change of the pump power we calculated the change of the waist radius on Nd:YAG and the stability criterion of the o- and e-ray two higher order modes based on the corresponding formula of the fundamental mode. The stability of laser resonators can be expressed by the stability criterion derived from the transfer matrix of the resonator [17]. Figure 5 plots the corresponding stability criterion (solid line) for both o- and e-ray as well as the mode radii (dotted line) as a function of incident pump power, respectively.

 

Fig. 5 Transverse sizes of both o- and e-ray in the laser crystal and their stability conditions as the functions of pump power

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As seen in Fig. 5, with the increase of pump power the thermal lensing effect changes the stability conditions and the transvers mode sizes of the o- and e-ray components. At pump power more than 4.3 W the resonator stability condition for o-ray is firstly satisfied (the blue solid line), and with pump power increase the mode radii (the blue dotted line) rapidly decreases and matches the pump beam waist radius r_P. But the resonator stability condition of e-ray does not satisfy (the red dotted line). Therefore the laser outputs only o-ray at pump power lower than about 4.7 W. As the pump power continues to increase the resonator stability condition of e-ray is starting to satisfy and the mode radii of e-ray (the red solid line) decreases rapidly to match with pump beam r_P with P_abs increase as well. While the stability conditions and mode radii of o-ray still maintains well. At this time the resonator stability conditions of both rays satisfy and their mode radius match r_P simultaneously. So both rays are outputted at the same time and form a complex pattern. When the pump power exceeds 6.2W, the resonator stability conditions of o-ray are not satisfy anymore, and the mode radii of o-ray rapidly increases and does not match r_P. While the stability conditions and mode radii of the e-ray are still maintained well. Therefore, the laser outputted only e-ray at this time. The result of the analysis clearly shows that the thermal lensing effect produced by pump power caused the difference in the stability conditions and the transvers mode sizes of these two components, therefore, the oscillation of the o-ray and e-ray can be obtained in different pump power regions. This result is consistent with that above experiment. To the best of our knowledge conversion between the radial and the azimuthal polarization in the intra-cavity way laser by inputting different pump power, Fig. 4 is reported for the first time. So if we can effectively control the thermal effect in the crystal and we can achieve higher output power level of the two vector beams in a laser.

In addition, from experiments we knew that the output power of the AP or RP was affected by many factors. By carefully adjusting the maximum output of the AP and RP were obtained at P_abs = 6.78 W. The mode stability of the AP and RP beam obtained in this experiment was very high. For AP, the laser output power reached maximum 2.4 W with o-o efficiency of 35.4%. For the sake of verifying the polarization degradation, the polarization purities of corresponding AP laser beam was measured by using the similar method as described in [19]. It showed an estimated value of 97.3% of the polarization degree. The M² factor of this laser mode was measured to be nearly M²_x = 2.36, M²_y = 2.23. For RP, the laser output power reached maximum 2.52 W with an o-o efficiency of 37.2%, the estimate value of polarization degree was 95.8%, the M² factor was measured to be nearly M²_x = 2.43, M²_y = 2.33.

4. Conclusions

In summary, the azimuthally and the radially polarized laser beams have been generated from the output of a Nd:YAG laser by using a birefringent crystal c-cut YVO₄. We can use a simple way by regulating inputting pump power to easily implement the conversion between the radially polarization and the azimuthal polarization in a laser. The oscillation was very stable. This approach for generating azimuthally and radially polarized beams offers many attractions over existing techniques and is well-suited to achieve azimuthally and radially polarized beams in a laser for other wavelength. Such polarization switching way in a laser was reproducible and would facilitate many applications. Endeavors to optimize the laser parameters are in progress.