1. Introduction

Cylindrical vector (CV) beams with spatially variant polarization have attracted widespread attention because spatial polarizations with specific distributions engender beam characteristics that can improve the usability of optical systems in various applications. Examples of these applications include particle acceleration, optical trapping, high-resolution microscopy, and material processing [1,2]. Typical CV beams are radially or azimuthally polarized. Existing methods for generating these cylindrical beams can be classified as passive or active methods. Passive methods involve varying the polarization state outside the laser cavity [3–5], whereas active methods provide direct lasing of CV beams by means of intracavity elements usually [6–14].

Recently, a CV beam was generated directly from a conventional laser without using an extra intracavity element. This beam can be classified as an active type owing to the direct generation of CV beams. It can be achieved via two main approaches: the first is to use a pump or gain profile to excite the CV beams; the second involves using a birefringent laser crystal as a distinguishing mechanism for identifying ordinary (o-ray) or extraordinary (e-ray) rays. In a previous study that employed an approach involving shaping the pump profile [11], laser diodes were arranged around the rod at the same angle to one another to achieve a cylindrically symmetric gain distribution and a radially polarized beam. For a microchip Nd:YVO₄ laser, radial polarization was realized by shaping the pump profile to achieve a ring distribution [12]. Furthermore, by employing pump reshaping and a thermal gradient across the crystal surface to control both the intensity and polarization profile, the output mode yielded a radially polarized beam for a monolithic microchip laser [13]. In addition, a conical gain profile was used for a hemispherical cavity to generate vector Bessel–Gaussian beams [14].

When a birefringent laser crystal was considered, the presence of birefringence led to o- and e-rays with different equivalent crystal lengths. This difference induced a small shift in the stable region of the e-ray. Consequently, although major portions of the stable regions of the o- and e-rays overlapped, the edges around this overlapped stable region only contained either o-rays or e-rays, i.e., one type of CV beam could be obtained [15–18]. Moreover, radially and azimuthally polarized beams can be generated by employing thermally induced birefringence and the thermal lensing effects of a Nd:YAG laser rod [15,16]. In hemispherical cavity configurations, c-cut Nd:YVO₄ or Nd:GdVO₄ lasers generate radially polarized beam oscillations because the positive birefringence of the crystal generates a larger expansion of the cavity length for an extraordinary ray [17]. An azimuthally polarized beam was generated from passively Q-switched Nd:GdVO₄ lasers using a cavity configuration that enabled operation around the edge of the stable region [18]. Although shaping the pump distribution has been considered, birefringence typically plays a critical role in generating CV beams.

The aforementioned reports [11,17,18] indicated that one type of CV beam can be supported at one specific edge of the stable region. It is possible that a large divergent angle and low beam quality, that is, a high M-square value, exist because high-order transverse modes can be induced when lasers are operated around the edge of the stable region. Moreover, cavities require fine tuning to distinguish between the power extraction efficiency of multiple transverse modes and to obtain the CV beam. In other words, the tunable region for the cavity to generate CV beams is small. Although a CV laser system employing thermally induced birefringence and thermal lensing effects is capable of obtaining a beam with a low M-square value [15,16], the system requires high pump power to induce birefringence and thermal lensing effects and the stability of CV beam depends on the pump power. Beams with low M-square values feature stronger focusing capabilities and afford longer working distances, which improve their applicability. Therefore, it is interesting to determine whether the conventional cavity configuration can self-sustain a high-quality CV beam in circumstances extending beyond the edge of the stable region.

In this paper, a selection rule is proposed for cavity configurations, with the aim of generating a CV beam with a low M-square value using a birefringent laser crystal. A four-element cavity was used to verify the selection rule through simulations and experiments, where the position of one of the end mirrors representing the output coupler was set as the tuning parameter. All the stable regions of the tuning parameters support CV beams, provided that the cavity configuration satisfies the selection rule. This cavity configuration prevents the laser from being operated at the boundary of the stable region, and thereby high-quality CV beams are easily excited in the experiments. Moreover, radially or azimuthally polarized beams can be obtained by varying the tuning parameter. This feature indicates the suitability of such cavities in polarization-dependent applications, where efficient azimuthal and radial polarizations are required for drilling holes in different materials [19].

2. Analyses and numerical simulations for cavity design

Typically, to analyze the stability of a multi-element paraxial cavity, the ABCD law and q-parameter are used [20, 21]. Considering an end-pump cavity configuration with a birefringent gain medium, Fig. 1 presents a schematic of a multi-element cavity configuration. M₁ and M₂ are the end mirrors of the cavity and have radii of curvature R₁ and R₂, respectively. The gain medium has a length of ${\ell _g}$ and is located close to M₁. The distance between the medium and the first intracavity element, E₁, is ${\ell _i}$, and the distance between the last intracavity element, E_n, and M₂ is ${z_f}$. When the transfer matrix from M₁ to M₂ (without M₁ and M₂) is computed to be $\left[ {\begin{array}{{cc}} a&b\\ c&d \end{array}} \right]$, the condition for cavity stability can be represented as

 

Fig. 1. Equivalent cavity considering the gain medium. G is the gain medium with length ${\ell _g}$. The transfer matrix is computed from arriving at the first intracavity element (E₁) to leaving the last intracavity element (E_n).

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To simplify the cavity design, we set M₁ and M₂ as planar end mirrors. The benefit of using planar end mirrors is that their corresponding radii of curvature (R₁ and R₂ in this case) are equal to infinity. As a result, the generalized g-parameters can be simplified to

In this work, the simulations and experiments focus on a four-element cavity with planar end mirrors M₁ and M₂, as shown in Fig. 2. Such a four-element cavity is typically used in mode-locked lasers. Two lenses, L₁ and L₂, are inserted sequentially in the cavity, and M₂ is chosen as the OC. The distance between L₁ and L₂ is z₁₂. Thus, the generalized g-parameters are

 

Fig. 2. Four-element cavity with end mirrors M₁ and M₂ and intracavity lenses L₁ and L₂. G is the gain medium with length ${\ell _g}$_, and M₂ is the OC.

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For the G₁-G₂ coordinate system, the trajectories of $({G_1},{G_2})$ with respect to varying ${z_f}$ for different values of ${\ell _i}$ are illustrated in Fig. 3. As ${z_f}$ = 0, the computed $({G_1},{G_2})$ is located at a position labeled as “${z_f}$ = 0”, and the next or right-hand $({G_1},{G_2})$ point corresponds to a 1-mm increment in ${z_f}$, i.e., the $({G_1},{G_2})$ coordinate for ${z_f}$ = 1 mm. By sequentially increasing ${z_f}$ in 1-mm increments, the trajectory of $({G_1},{G_2})$ corresponding to the increase ${z_f}$ is obtained, where the red diamonds and blue circles represent the $({G_1},{G_2})$ values of the o- and e-rays, respectively. The solid diamonds and circles indicate that the $({G_1},{G_2})$ values located in the stable regions with $0 \le {G_1}{G_2} \le 1$. From Eq. (4), ${G_2}$ is independent of ${z_f}$ and ${G_1}$ is linear to varying ${z_f}$. Thus, the trajectory of $({G_1},{G_2})$ shows a horizontal line on varying ${z_f}$. Moreover, for a specific ${\ell _i}$, a birefringent medium results in different values of ${z_i}$ for the o- and e-rays and ${G_{2,o}} \ne {G_{2,e}}$, where the suffixes “o” and “e” indicate the parameters representing the o- and e-rays, respectively. Two horizontal lines for the o- and e-rays are formed on varying ${z_f}$ in the ${G_1}$-${G_2}$ coordinate system, as shown in Fig. 3. The distance between two horizontal trajectories is ${G_{2,o}} - {G_{2,e}} = \textrm{0}\textrm{.13}$ and two lines simultaneously shift as varying ${\ell _i}$.

 

Fig. 3. The trajectories of (G₁,G₂) by varying ${z_f}$ for various ${\ell _i}$. values of (a) ${\ell _i}$. = 79.5 mm, (b) ${\ell _i}$=80.5 mm, and (c) ${\ell _i}$ = 81.5 mm. The gray region with 0$\; \le \textrm{\; }$G₁G₂$\; \; \le \textrm{\; }$1 represent the stable region. As ${z_f}$ = 0, the computed (G1,G2) is located at the position labelled by “${z_f}$ = 0”, and the next or right-hand (G1,G2) point corresponds to a 1-mm increment of ${z_f}$.

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In Fig. 3(a), where ${\ell _i}$ = 79.5 mm, both horizontal trajectories are below the horizontal axis (${G_2}$ = 0) and the stable region of the o-ray is greater than that of the e-ray. The region labeled “AP_b” has ${G_{1,o}}{G_{2,o}} < 1$ and ${G_{1,e}}{G_{2,e}} > 1$ and indicates that the o-ray is stable but the e-ray is unstable. The cavity configuration was more suited to an azimuthally polarized beam because the polarization of the o-ray is located on the sagittal plane. Thus, AP_b represents the region that supports azimuthally polarized beams, and the subscript “b” indicates the cavity configuration for which the laser is operated around the G₁G₂ = 1 boundary. In contrast to AP_b, the region labeled “RP_b,” as shown in Fig. 3(c) with ${\ell _i} = $ 81.5 mm, has ${G_{1,o}}{G_{2,o}} > 1$ and ${G_{1,e}}{G_{2,e}} < 1$. The “RP_b” region can support a radially polarized beam. These G₁G₂ conditions around the G₁G₂ = 1 boundary are a typical cavity design adopted in many previous studies. In the experiments, it was possible to excite high-order transverse modes at the boundary of the stable region and even slightly outside the stable region. For example, the high-order transverse modes of the e-ray mix with the fundamental mode of the o-ray in the AP_b region to destroy the azimuthal polarization. Consequently, high-order transverse azimuthally polarized modes are generated around or slightly outside the boundary of the stable region of the o-ray, owing to the lack of high-order transverse modes of the e-ray. Typically, an aperture is inserted to suppress the high-order modes and improve the beam quality during the experiments.

It is interesting that for a specific ${\ell _i}$, such as when ${\ell _i}\; $ = 80.5 mm as in Fig. 3(b), the $({G_1},{G_2})$ trajectory of the o-ray lies above the horizontal axis, whereas the trajectory of the e-ray lies below the horizontal axis. In all stable regions, only one ray survives at each ${z_f}$. In the region to the left of the vertical axis (labeled RP_s), only the e-ray is sufficiently stable to sustain a radially polarized beam. Similarly, an azimuthally polarized beam exists in the region to the right of the vertical axis (labeled AP_s). The subscript “s” indicates that that all the stable regions can support CV beams. This condition, having one trajectory above the horizontal axis and the other trajectory below the horizontal axis, corresponds to ${G_{2,o}} > 0$ and ${G_{2,e}} < 0$. Based on ${G_{1,o}} = {G_{1,e}}$ at a fixed ${z_f}$, the condition ensures that at least one of the $({G_{1,o}},{G_{2,o}})$ and $({G_{1,e}},{G_{2,e}})$ coordinates is located in the second or fourth quadrant, i.e., in the unstable regions for every ${z_f}$. This is because the stable region with $0 \le {G_1}{G_2} \le 1$ lies in the first and third quadrants. This condition provides a criterion to confirm that one type of CV beam will be suppressed. The other type of CV beam will survive if the configuration satisfies $0 \le {G_1}{G_2} \le 1$. Thus, we can define the selection rule as

Figure 4 displays the ${z_f}$-dependent spot sizes for M₁, which the cavity is unstable if the spot size is not real or nonexistent. As shown in Fig. 4(a), where ${\ell _i}$ = 79.5 mm, the cavity is stable for the o-ray and unstable for the e-ray at 14.49 mm< ${z_f}$ < 18.14 mm, which is associated with the AP_b region. However, as shown in Fig. 4(b), where ${\ell _i}$ = 80.5 mm, there is no overlap between the stable regions of the e- and o-rays; the e-ray is stable at ${z_f}$ $\le $ 26.04 mm (the RP_s region) and the o-ray is stable at ${z_f}$ > 26.04 mm (the AP_s region). A wide ${z_f}$-tuning region can generate CV beams. Summarizing the stable regions for various ${\ell _i}$ values, Fig. 5 demonstrates the stable region with respect to ${\ell _i}$ and ${z_f}$. The stable region of the e-ray shifts $({\ell _g}/{n_o} - {\ell _g}/{n_e})$ in the horizontal direction. This shift also corresponds to the size of the preferred region. The red and blue regions correspond to azimuthal and radial polarizations, respectively. In the gray region, the o- and e-rays can both be excited to lead to typical polarization or non-CV polarization. The AP_s and RP_s regions are identified in the preferred$\; $range of $\textrm{80}\textrm{.35\; mm} < {\ell _i} < \textrm{80}\textrm{.74\; mm}$. In the ${z_f}$ tunable regions, the ranges of AP_b and RP_b decrease as the distance between ${\ell _i}$ and the preferred region increases. It is also beneficial to form low-order CV beams in the AP_b or RP_b regions around the preferred region; however, only one type of CV beam can be obtained at a specific ${\ell _i}$. In practice, the preferred region also exists when the gain medium is not located near M₁, such as the gain medium position between L₁ and L₂ in Fig. 2. However, the closed form is more complicated, and the numerical calculation for determining the preferred region must be used appropriately.

 

Fig. 4. Spot sizes for M₁ as a function of ${z_f}$ at (a)$\; {\ell _i}$. = 79.5 mm and (b) ${\ell _i}$=80.5 mm.

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Fig. 5. Stable region with respect to ${\ell _i}$ and ${z_f}$. “AP” and “RP” denote the preferred regions for azimuthally and radially polarized beams, respectively. The subscript “b” indicates the cavity configuration for which the laser operates around the G₁G₂ = 1 boundary, and the subscript “s” indicates that all stable regions support CV beams at a specific ${\ell _i}$. The gray region represents the cavity configuration supporting typical polarization or non-CV polarization.

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From Eq. (4), the existence of the selection rule requires that $c^{\prime} \ne 0$. As such, the two-mirror cavity cannot support the selection rule because its $\left[ {\begin{array}{{cc}} {a^{\prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}} \end{array}} \right]$ matrix is regarded as a unit matrix, i.e., $c^{\prime} = 0$. The simplest cavity configuration that satisfies the selection rule is a three-element cavity involving two end mirrors with an intracavity lens. In this case, $c^{\prime} ={-} 1/{f_i}$ and ${f_i}$ is the focal length of the intracavity lens. If the intracavity lens is replaced by thermal lensing of the birefringent gain medium, the selection rule may be satisfied for a two-mirror cavity [16]. However, $c^{\prime}$ depends on the pump power. The selection rule is also applicable to an isotropic or weakly birefringent gain medium if an additional birefringent crystal is considered, as in [6]. This is because a gain medium incorporating a birefringent crystal can contribute to the difference between the equivalent distance, ${z_i}$, of o- and e-rays. Moreover, when curved end mirrors are considered, the horizontal trajectory of $({G_1},{G_2})$ for two planar end mirrors is transformed into an oblique line. The slope of this oblique line increases as the radii of curvature, R₁ and R₂, decrease. Thus, the trajectory passes through three quadrants to reduce the tunable region of ${z_f}$ for generating a CV beam in the preferred region.

3. Experimental results

For the experiments, a four-element Nd:GdVO₄ laser was chosen to verify the abovementioned simulations, and the cavity parameters used in the experiments were the same as those used in the simulations. A diode laser with a wavelength of 808 nm was focused onto the Nd:GdVO₄ crystal and served as the pump source. The planar end mirror M₂ serving as the OC exhibited a reflectivity of 90%. Figure 6 demonstrates the stable regions in response to varying ${\ell _i}$ and ${z_f}$, where the red and blue regions represent the azimuthally polarized and radially polarized regions, respectively. The gray region indicates that the laser is not a CV beam and has typical polarization-related characteristics in the c-cut laser crystal. The experimental results agree with the numerical simulations as shown in Fig. 5.

 

Fig. 6. Stable region with respect to ${\ell _i}$ and ${z_f}$ in the experiments, compared to the simulation results in Fig. 6.

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From Eq. (7), the preferred CV operation region corresponds to$\; \textrm{80}\textrm{.6\; mm} \le {\ell _i} \le \textrm{81}\textrm{.2\; mm}$. Figure 7 shows the patterns for varying ${z_f}$ at ${\ell _i}$=81.0 mm. In the preferred region, the sequential polarization transformations are radial, non-CV, and azimuthal for increasing ${z_f}$, that is, in response to moving the OC or M₂. Ring patterns with radial polarization are observed at ${z_f} \le \textrm{20\; mm}$, which is transformed into a complex transverse pattern as $21\textrm{\; mm} \le {z_f} \le \textrm{27\; mm}$. For ${z_f} \ge \textrm{28\; mm}$, the pattern once again forms a ring shape but with azimuthal polarization. The polarization of a CV beam can be verified by rotating the polarizer angle to observe the patterns. For example, Fig. 8 shows the transverse patterns obtained at various polarizer angles for ${z_f}$ = 14 mm and 30 mm. The arrows represent the direction of the polarizer, and “N” denotes the original pattern for which a polarizer is not used. At ${z_f}$ = 14 mm, the intensities in the direction vertical to the polarizer direction are suppressed, in which the resulting beam features radial polarization. In contrast, azimuthal polarization is observed at ${z_f}$ = 30 mm. Moreover, the degrees of polarization, measured based on a method used in [9], are $90.17\; \pm \; 3.02\%$ and $\textrm{89}.06\; \pm \; 4.30\%$ for ${z_f}$ = 14 mm and 30 mm, respectively, which indicates good polarization performance.

 

Fig. 7. Output intensity patterns with respect to varying ${z_f}$ at ${\ell _i}$=81.0 mm. Beams are radially and azimuthally polarized for ${z_f} \le 20\; \textrm{mm}$ and ${z_f} \ge 28\; \textrm{mm}$, respectively.

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Fig. 8. (a) Intensity pattern under various polarized conditions at ${z_f}$ = 14 mm (top column) and 30 mm (bottom column) for ${\ell _i}$=81.0 mm. The arrow represents the orientation of the polarizer, and “N” represents the original pattern obtained without adding a polarizer. (b) The y-axis intensity distributions at ${z_f}$ = 14 mm (top column) and 30 mm (bottom column) correspond to the yellow dashed lines labeled in (a). The red lines are the fitting profiles based on HG₀₁.

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Based on a CV beam decomposed by the sum of the Hermite–Gaussian modes with orthogonal polarizations, the fundamental or lowest-order modes of the radially polarized beam, $\overrightarrow {{E_r}} $, and azimuthally polarized beam, $\overrightarrow {{E_\phi }} $, can be represented as [1]

To thoroughly evaluate the beam quality in the preferred CV operation region, the M-square (${M^2}$) value of the beam was measured [22]. ${M^2}$ is the beam quality factor, which is defined as the ratio of the space-bandwidth product of the real beam to that of the fundamental Gaussian beam. The space-bandwidth product represents the product of the minimum spatial variance at the beam waist and the variance in spatial-frequency. If a stable laser cavity oscillates simultaneously in multiple Hermite–Gaussian modes ($H{G_{pm}}$) with a relative amplitude ${c_{pm}}$, the beam quality factors are $M_x^2 = \mathop \sum \limits_{p = 0}^\infty \mathop \sum \limits_{m = 0}^\infty ({2p + 1} ){|{{c_{pm}}} |^2}$ and $M_y^2 = \mathop \sum \limits_{p = 0}^\infty \mathop \sum \limits_{m = 0}^\infty ({2m + 1} ){|{{c_{pm}}} |^2}$ [22]. Therefore, $M_x^2$ and $M_y^2$ represent the ${M^2}$ values based on the variances computed in the x- and y-directions, respectively. When the fundamental mode of a radially polarized beam passes through a horizontal polarizer, $H{G_{10}}$ is retained and the beam quality factors are $M_x^2 = 3$ and $M_y^2 = 1$. Based on the same analysis, the fundamental mode of the radially polarized beam has quality factors of $M_x^2 = 1$ and $M_y^2 = 3$ for vertical polarization. The fundamental mode of the azimuthally polarized beam has x-polarized beam quality factors of $M_x^2 = 1$ and $M_y^2 = 3$ and y-polarized beam quality factors of $M_x^2 = 3$ and $M_y^2 = 1$.

Figures 9(a) and 9(b) depict $M_x^2$ and $M_y^2$ as a function of ${z_f}$ for the horizontal and vertical polarizations, respectively. At ${z_f} \ge 28\; \textrm{mm}$, the azimuthally polarized beam operates almost in the fundamental mode; for example, $M_x^2 = 1.18\; \pm \; 0.10$ and $M_y^2 = 3.16\; \pm \; 0.08$ for the x-direction polarization and $M_x^2 = 3.08\; \pm \; 0.10$ and $M_y^2 = 1.14 \pm \; 0.04$ for the y-direction polarization as ${z_f} = 30\; \textrm{mm}$. The fundamental mode of the radially polarized beam is 12 mm$\le {z_f} \le $18 mm, and the low-order mode of the radially polarized beam corresponds to ${z_f} < 12\;\textrm{mm}$. These low-order modes are dominated by HG₁₀ and HG₀₁ and mixed with a few HG₂₀ and HG₀₂ modes. This is because the M-square values are close to those of HG₁₀ or HG₀₁. Moreover, the intensities of the outside ring patterns, as shown in Fig. 7, are less than those of the fundamental modes. In practice, the experimental results were achieved solely by tuning the position of the OC, without considering the overlap integral or mode matching between the intensity distributions of the pump beam and the intracavity mode. The pump spot size was approximately 380 μm, and the cavity spot sizes at M₁ were less than 20 μm, as shown in Fig. 4(b). Although the condition of mode matching is subpar, most of the cavity configurations exhibit strong energy-extraction capabilities to self-sustain the fundamental mode. Further optimization of the pump conditions at each ${z_f}$ can refine the beam quality, slope efficiency, and operating ranges of the fundamental CV beams. The experimental results verified that fundamental CV beams were generated in the preferred region and that these beams exhibited a widely tunable operating range. Moreover, fundamental modes related to azimuthal polarization or those related to radial polarization could be switched simply by moving the OC.

 

Fig. 9. M-square values as a function of ${z_f}$ for x- and y-direction polarizations at ${\ell _i}$ = 81.0 mm.

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In Figs. 7 and 9, the complex transverse pattern with high ${M^2}$ results from a value of ${G_1}{G_2}$ around zero as $21\; \textrm{mm} \le {z_f} \le 27\; \textrm{mm}$. Because ${G_{2,o}}$ and ${G_{2,e}}$ are independent on ${z_f}$ from Eq. (4), ${G_{1,o}}{G_{2,o}}$ linearly increases and ${G_{1,e}}{G_{2,e}}$ linearly decreases as in accordance with increases ${z_f}$. ${G_{1,o}}{G_{2,o}}$ and ${G_{1,e}}{G_{2,e}}$ simultaneously equal zero as ${z_f} ={-} a^{\prime}/c^{\prime}$. When a cavity configuration approaches the condition of ${G_1}{G_2} = 0$, it is expected to generate multiple and high-order transverse modes. The number and order of the generating transverse modes could be different for the e- and o-rays. Thus, not only are the patterns complicated, but the CV polarizations are broken. Moreover, the complex transverse patterns differ as the beam passing through the horizontal and vertical polarizers for some ${z_f}$. This means that the polarizations are spatially dependent but not CV-polarization, and it is worth exploring this aspect in future research.

4. Conclusions

In summary, the capability of conventional cavity configurations in generating CV beams was discussed with consideration of the birefringence of the gain medium. By analyzing the generalized g-parameters, a selection rule to determine the cavity configuration was proposed, with the aim of directly generating fundamental CV laser beams. Cavities that satisfy the selection rule offer two distinct advantages. First, all stable regions for the tuning parameter can support CV beams; therefore, the fundamental or lowest-order mode operation excludes the boundary of the stable region. Second, radially or azimuthally polarized beams can be obtained simply by moving the OC. Based on measurements of the beam quality factor and polarization characteristics, it is evident that a wide tuning range in the stable region promotes the formation of fundamental or at least low-order CV beams. The experimental results corroborate the analytical and simulation results.