1. Introduction

Radially and azimuthally polarized vector beams are now among the more well-known cylindrical vector beam categories. Cylindrical vector beams demonstrating cylindrically symmetric polarization have attracted considerable interest because of their potential applications in electron acceleration [1], optical trapping and manipulating [2], and material processing [3]. Two procedures, known as the passive and active methods, generate radially and azimuthally polarized beams [4]. The passive method is used to modify the beam outside the laser cavity by employing an interference process [5] and twisted nematic liquid crystal [6,7]. Using an intracavity optical element, the active method directly produces the cylindrical vector beam. These active-type lasers were developed by using birefringence elements [8,9], discontinuous phase elements [10], and thermally induced bipolar lensing [11], as well as by applying a photonic crystal grating as a polarization-selective output coupler [12] and by placing a Brewster-angle axicon inside the cavity [13–15].

Recently, using the characteristics of the gain medium to distinguish extraordinary and ordinary rays could obtain the cylindrical vector beam [11,16]. Because the strong birefringence of the Nd:YVO₄ crystal could generate the optical path difference between extraordinary and ordinary rays, the cavity design enabled the extraordinary ray to become stable only to achieve radial polarization [16]. By contrast, an isotropic Nd:YAG crystal lacks birefringence. Based on thermal lensing and thermally induced birefringence to generate the different focusing between the radial and tangential polarizations, Nd:YAG laser oscillators were developed to produce low-loss stable oscillation in radial or azimuthal polarizations [11]. In both of the mentioned cases, an intracavity aperture could be used to enhance the contrast ratio of the polarization.

Moreover, an intracavity axicon has been used to generate a radially polarized laser output since 1972 [13]. Mushiake et al. first used a Brewster-angle axicon mounted in the subresonator to produce a radially polarized He-Ne laser at 0.633 μm [13]. Later, Kozawa and Sato designed a new Brewster optical element consisting of convex and concave conical prisms to generate a radially polarized beam in a plane-concave Nd:YAG laser [14]. These two methods provide a paraxial mode output, and involve using multiple coatings at the conical surface to enhance the polarization selection. In contrast to previous studies, Bisson et al. obtained off-axis, thin ring-like and arc-like radially polarized laser beams by using an intracavity axicon close to the Brewster angle [15]. The applications discussed in their paper indicated that the generation of RP beams in the form of a narrow ring could be useful for irradiating targets by using p polarized light near these maxima of absorption, providing higher energy transfer to metals and plasma heating that is more efficient. A Brewster-angle axicon is widely used to generate a radially polarized laser beam. However, this axicon is detrimental to the azimuthally polarized beam, as represented by the reflective loss of azimuthal polarization; however, there is nearly no loss of radial polarization through the Brewster-cut interface.

This paper proposes that a large-apex-angle axicon can be used to achieve an off-axis laser with radial or azimuthal polarization by configuring the cavity length only. In contrast to previous studies, the apex angle of the axicon was far from the Brewster angle and could be an arbitrary angle in practice. A large-apex-angle axicon supports a cylindrical symmetrical mechanism, and expands the difference between ordinary and extraordinary rays to generate off-axis cylindrical vector beams. Moreover, a Brewster-angle axicon is unfavorable for supporting azimuthally polarized beams, and this shortcoming is overcome in the large-apex-angle axicon system.

2. Off-axis beams with radial and azimuthal polarizations

Figure 1(a) schematically depicts the experimental setup of the Nd:YVO₄ laser with an intracavity axicon. A diode laser with a wavelength of 808 nm and a maximum output power of 30 W was collimated to a parallel-like beam, and focused onto the Nd:YVO₄ crystal by using another 10$\times$ objective lens. The c-cut Nd:YVO₄ crystal had dimensions of 3$\times$3$\times$1 mm³ and a doping concentration of 1%. One side of the Nd:YVO₄ crystal displayed an antireflection coating at 808 nm and a high reflection coating at 1064 nm. This side also acted as an end mirror of the laser cavity. The other side exhibited an anti-reflection coating at 1064 nm to reduce the effect of the intracavity etalon. A concave mirror with a radius of curvature of R_c = 80 mm and a reflectivity of 94% was used as the other end mirror and the output coupler. The axicon was manufactured using fuse silica, with an optical index of n = 1.45 and the apex of α = 178°. The distance between the laser crystal and axicon was labeled as z_a. The output power and beam pattern were measured using a power meter and a CCD camera, respectively.

 

Fig. 1 (a) Experimental setup (b) Output power as a function of pump power at z = 3.6cm.

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The off-axis laser beam was sustained using the multiple-pass mode proposed by Wu [17], and this mode was also observed in the radially polarized laser with an intracavity Brewster-angle axicon [15]. The off-axis mode mainly occurred at the boundary of the stable region. Thus, the cavity length, L, was approximately 80 mm, indicating that the cavity had nearly a hemispherical configuration. When L increased at z_a = 3.6 cm, the polarizations of the laser were sequenced into non-polarization, azimuthal polarization, and radial polarization. Although the cavity length was varied, the slope efficiencies and thresholds for azimuthal polarization were approximately 20.5% and 0.20 W, respectively, as shown in Fig. 1(b). Figure 2 displays the pattern formations of the azimuthally polarized laser at various polarized directions, in which the polarized patterns were pictured by passing them through a linear polarizer, and the red arrow represents the direction of the polarization angle. Figure 2(a) illustrates the original pattern without adding a polarizer to the multi-lobe ring pattern. Adding the polarizer caused the part of the pattern corresponding to the parallel direction of the polarization angle to disappear, as shown in Figs. 2(b)–2(d). Consequently, an azimuthally polarized beam was formed. The azimuthal distribution of ring intensity was not uniform, which could have partly resulted from the pump laser having a degree of polarization of 0.22 and the thermally induced asymmetry in the crystal affecting the extraction efficiency.

 

Fig. 2 (a)-(d) The near-field pattern at various polarized angle. “N” represents the pattern measured without adding polarizer and the red arrow represents the direction of the polarization angle, which are the same representations in the following figures.

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Furthermore, an intracavity knife edge was inserted to block a part of the pattern for exploring the off-axis characteristics, as shown in Fig. 3. Figure 3(a) depicts the blocked part of the knife edge with the dashed green shape. Figure 3(b) depicts not only the blocked part but also the central symmetrical portion that vanished in the laser pattern. The same result occurred when the inserted angle was varied. Thus, the lobes located at the azimuthal angles of θ and θ + π were a spot pair of a two-pass mode. The ring pattern was formed by multiple pairs at various azimuthal angles. Because each spot pair had the same divergent angle and was focused on the same focal plane, the far-field pattern was displayed as a spot, as shown in Fig. 4. Figures 4(a) and 4(b) show the far-field patterns and intensity distributions with vertical and horizontal polarizations, respectively. The intensity was normalized to the maximal intensity in the vertical direction. The far-field image demonstrates a spot with smoothened intensity distribution, and partial polarization resulted from the non-uniform pattern distribution.

 

Fig. 3 (a) The blocked part of the knife edge with the dashed green shape, and (b) the pattern after blocked.

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Fig. 4 (a) and (b) show the far-field patterns and intensity distributions at horizontal and vertical polarization.

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Figure 5 shows the polarization characteristics of the azimuthally polarized laser. A 300-μm-wide slit and a polarizer, both mounted on a rotation stage, were used to measure the polarization at the radial part of the laser beam. After using the slit to determine the measured portion of the pattern, the beam passed through the polarizer, and the power was then measured by using the power meter. The horizontal direction was set at 0° for both orientations of the slit and the polarizer angle. The transmitted power as a function of the polarizer angle is presented in Fig. 5(a). The orientation of the slit is −60°. The red line is the fitting curve with the sin² function, and the curve indicates the polarization direction and degree of polarization. The degree of polarization is defined as (I_max-I_min)/(I_max + I_min), where I_max and I_min are the maximal and minimal intensity of the fitting curve, respectively. The yellow line in the inset of the figure represents the orientation of the slit. The polarization direction was 30°, and the degree of polarization was 96.6% in this case. The polarization direction and degree of polarization determined by varying the orientation of the slit are shown in Fig. 5(b). The slope of 0.99 ± 0.006 with a shift of 90° from the red fitting curve indicates favorable characteristics of azimuthal polarization. The degree of polarization is 95.4% ± 2.6%.

 

Fig. 5 (a) The output power as a function of the angle of the analyzer for the ring-pattern laser beam as the slit angle of - 60°. (b) Polarization direction as a function of the slit angle

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Tuning the output coupler to increase the cavity length causes the polarization to transfer from azimuthal to radial polarization. Figure 6 displays the radially polarized pattern at various polarization angles. The portion of the pattern, which is perpendicular to the direction of the polarizer, disappeared. After measuring and computing polarization characteristics, the degree of polarization was 94% ± 3.7% for various slit angles, and the slope in the polarization direction versus the slit angle was 0.98 ± 0.004.

 

Fig. 6 The ring-pattern beam with radial polarization at various polarization angles.

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Moreover, when the axicon was moved toward the output coupler to z_a > 4.45cm, the pattern transformed from a ring to an arc around the hemispherical cavity configuration. At a fixed z_a in z_a > 4.45cm, the polarizations of the laser were also sequenced into non-polarization, azimuthal polarization, and radial polarization while increasing the cavity length. In contrast to the transform in the ring pattern, the arc beams must slightly adjust the axicon to optimize the polarization contrast usually. Figures 7(a) and 7(b) respectively display radially and azimuthally polarized arc patterns at different polarizations that were generated at z_a = 5 cm. The degree of polarization reached an average of 97% for the arc-pattern beam, a value that was higher than that of the ring-pattern beam. The slope efficiency reduced to 15.9%, and the threshold enhanced to 0.27 W. Based on the mentioned results, the distance between the crystal and axicon z_a dominates the type of pattern, and the cavity length determines the type of polarization when the cavity configuration operates around the hemispherical configuration. By increasing z_a, the pattern transfers from the ring to the arc and, at each pattern, the polarization varies from azimuthal to radial polarization by increasing the cavity length.

 

Fig. 7 The arc-pattern beams with (a) radial and (b) azimuthal polarization at various polarized angles.

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3. Analyzing the divergent angle of off-axis beams

Figure 8(a) is a geometrical diagram illustrating the tracings of ordinary and extraordinary rays while the cavity operated around the hemispherical configuration [18]. The thicknesses of the axicon and crystal were enlarged for determining the ray tracing in the figure. As shown in Fig. 8(a), when the ordinary ray propagates into the first plane of the axicon at Point A with an incident angle of θ, the oblique angle, θ_out, at Point B is derived to be

 

Fig. 8 (a) Geometrical diagram of ray tracing at hemispherical cavity. (b) The divergent angle as a function of the axicon’s position. The solid square and empty triangle represent the experimental results with β = 1° for the Nd:YVO₄ and Nd:YAG lasers, respectively. The different lines correspond to the numerical results for the various β’s values.

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Furthermore, an isotropic Nd:YAG crystal was substituted for the Nd:YVO₄ crystal to verify the contributions of the birefringence of the crystal and the axicon. The dimensions and coatings of the Nd:YAG crystal were the same those of the Nd:YVO₄ crystal. The pump power was 500 mW, in which the low pump intensity reduced the thermal lensing effect. The contrast ratios of the polarization in the Nd:YAG laser were lower than 5 for various cavity configurations. The results indicate that the generation of the radially and azimuthally polarized beams originates in the birefringence of the crystal. Regarding the contribution of the axicon, Fig. 8(b) shows a divergent angle as a function of z_a in Nd:YVO₄ and Nd:YAG lasers. In the first step of the setup procedure, the cavity length without the axicon was fixed at 8 cm, which was verified by measuring the longitudinal mode beating. The divergence angle was then measured by inserting and moving the axicon in the cavity. The divergent angle was independent of the laser crystal. Because the crystal was thin and the equivalent thickness defined as the actual thickness dividing the index could be approximated, the configuration in the simulation simply involved considering the hemispherical cavity with an intracavity axicon. The experimental results are consistent with the numerical results. Thus, the pattern formation was dominated by the axicon. The pattern transition from the ring to the arc could be a result of a break in symmetry in the hemispherical configuration, as observed by Wu [17]. This pattern-expansion characteristic also indicated that the increasing tendency of the divergent angle in the experiment was faster than that of the theoretical result in the region of the arc pattern.

Moreover, the divergent angle decreases as β decreases at a specific z_a, as shown in Fig. 8(b). When β approaches to zero, the effect of the conical surface disappears, and the divergent angle approaches to zero. In practice, in a hemispherical cavity without an axicon, all oblique rays from the center of hemisphere will normally incidence to curved mirror and repeat the ray path after two round trips by ray tracing. Thus, a complex pattern forms, as shown in Fig. 9. The beam shows the characteristics of the radial polarization. Based on the birefringence of the Nd:YVO₄ crystal, a radially polarized beam could be generated in the region, which was stable for the extraordinary beam and was unstable for the ordinary beam. By contrast, a uniquely incident angle exists in the cavity with an axicon to form a stable ring pattern. Because the pattern corresponds to the solution of the ray tracing, the intensity is as stable as that in stable configuration in experiments. In summary, the large-apex-angle axicon dominates and stabilizes the generation of ring or arc patterns, and enlarges the off-axis path difference between the ordinary and extraordinary rays to form a specifically polarized beam. Regarding the Brewster-angle axicon, the small apex angle quickly diverts the ray from the output coupler, and the Brewster cut becomes detrimental to azimuthal polarization.

 

Fig. 9 The pattern in hemispherical configuration without the axicon at various polarized angles.

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4. Conclusion

When a laser operates around the hemispherical configuration, a radially and azimuthally polarized laser operation with off-axis modes can be achieved using an intracavity large-apex-angle axicon in cavity configuration. The beam transfers from azimuthal to radial polarization only when the cavity length is increased. The degrees of polarization can be up to 94% ± 3.7% and 95.4% ± 2.6% for the radially and azimuthally polarized beams, respectively. Although the degree of polarization is lower than that of using a Brewster-angle axicon, the system still indicates excellent polarization characteristics.