1. Introduction

The controlled manipulation of light's spatial and polarization properties has acquired significant attention in recent years due to its versatile applications across a wide spectrum of scientific and technological domains. Vector beams hold promise for revolutionizing various fields from fundamental researches to future applications, including optical trapping and microscopy [1–3], advanced laser material processing [4–6], optical communications [7,8], advanced imaging [9,10], quantum optics [11], and quantum computing [12–14]. Efficient and precise generation of spatial light with polarization degree of freedom that stands as a pivotal and imperative challenge in the realm of technological advancement. Diverse methodologies and strategies are harnessed to regulate the spatial and polarization attributes of light, facilitating the creation and manipulation of vector beams. Prevalent techniques entail the use of spatial light modulators (SLMs) [15–17] and diffractive optical elements (DOEs) [18,19] to modulate the intensity and polarization profile of the incident beam. However, most of these methods require the integration of polarization optical components to achieve the polarization degree of freedom. An alternative method involves the deployment of q-plates, characterized by variable phase retardation and swift-axis orientation, yielding vector beams that exhibit polarization distributions varying across spatial dimensions [20–22].

In recent years, lasers have exhibited significant advancements in the realms of quantum communication and entanglement, underscoring the vital role of coherent and stable light sources in practical applications [23,24]. In our previous research, we discovered that by modulating the transverse and longitudinal mode coupling strengths within a laser's resonant cavity, it is possible to generate three-dimensional coherent states under specific cavity conditions. The experimental outcomes showcased the correlations between quantum and classical domains, albeit within the context of scalar optical fields [25,26]. To achieve vector optical fields with correlated polarization, a more complex integration of phase-controlling optical elements within the optical system is necessary, such as birefringence crystals [27], dichroic structures [28], and metamaterials [29], or utilizing materials exhibiting enhanced birefringent properties as gain media inside cavities [30–33]. The general theoretical work on multi-polarization tuning was also established [34]. Intra-cavity conversion can be attained by manipulating cavity length and pump power, capitalizing on the distinct stability criteria associated with ordinary and extraordinary rays [35]. Most polarized light is typically characterized using two mutually perpendicular bases. However, the polarized light discussed in this work includes elliptical polarization states, and its polarization angles are systematically analyzed. Our preceding investigations revealed that by exploiting geometric beam shaping within a spherical resonant cavity coupled with phase control of the birefringent crystal, the laser output could be manipulated to exhibit discrete circular and linear polarization states [36,37], though restricted to a few polarization geometries.

In this present study, we employ an efficient cylindrical resonant cavity configuration to achieve phase-controllable, fully polarized laser modes. Diverging from the earlier spherical resonant cavity design, the significant astigmatic properties of the cylindrical resonator and its strong interaction with birefringent gain media provide an increased range of stable polarization options in the laser output. Our experimental results demonstrate that employing off-axis excitation enables precise generation of distinct linearly polarized higher-order Hermite-Gaussian modes. Further manipulation of the resonator length facilitates the formation of multi-polarization laser beams with linear and elliptical polarizations. Numerical analyses of the phase characteristics exhibit excellent agreement with experimental findings, and the polarization states evolution of the laser output can be mapped onto the Poincaré sphere. The precise control of diverse polarization configurations in lasers holds promising application potential within the rapid growing field of quantum optics.

2. Experimental setup and results

In this study, we present the generation of laser modes utilizing a diode-pumped $\textrm{Nd}:\textrm{YV}{\textrm{O}_4}$ plano-concave cylindrical cavity as shown in Fig. 1(a). The front mirror of the laser cavity consists of a cylindrical mirror with a radius of curvature $R = 10\; \textrm{mm}$. The entrance face is coated with an anti-reflection (AR) coating specifically designed for 808 nm, while the opposite side is coated with a high-reflectivity (HR, > 99.8%) coating for 1064 nm. A c-cut 1.0-at. % $\textrm{Nd}:\textrm{YV}{\textrm{O}_4}$ crystal of 1 mm thickness is employed as the gain medium (GM). The crystal is coated with AR and HR coatings of 1064 nm on its entrance face and exit face, respectively. The refractive index is ${n_o} = 1.9573$ for ordinary ray and ${n_e} = 2.1652$ for extraordinary ray. The laser cavity is pumped by an 808 nm laser diode coupled with a fiber core diameter of 105 µm and a numerical aperture of 0.22, capable of delivering a maximum power of up to 6 W. Given the high absorbance of Nd:YVO₄ at 808 nm, we use an 808 nm laser diode as the pumping source to enhance the efficiency of laser emission [38,39]. The 808 nm light source is directed into the gain medium through a focusing lens, resulting in a spot size of 50 µm . The trajectory of the laser modes, spanning from the near field to the far field, is observed and recorded using an objective lens. The polarization characteristics of the laser modes are analyzed by employing a multi-order quarter-wave plate (QWP) and a polarizer (POL). Figure 1(b) presents the intensity and polarization state of HG modes ranging from (0,0) to (0,11) with an offsets of 10 µm for each order spacing, and the cavity length L is set to be 6.2 mm. As the offsets increase from 0 µm to 110 µm , a noticeable change in the polarization angle occurs, increasing from 48$^\circ $ to 72$^\circ $, as indicated by the colored arrows. Figure 1(c) depicts the evolution of the linear polarization angle with increasing HG mode order, revealing the presence of two distinct slopes in the polarization shift. Similarly, in Fig. 1(d), the divergence angle for different mode orders also exhibits two slopes, corresponding to the polarization change. These results indicate a strong connection between the divergence angle and the polarization states. The observation of the dual-slope phenomenon may attribute to the boundary condition of the cavity, stability of the laser, and the thermal effect by pumping source. To maintain a stable laser system in the cavity, a consistent optical path is crucial, ensuring coherence in both phase and polarization after each roundtrip. The HG modes exhibit a small divergence angle, concentrated near the pumping spot. Thermal effects on the gain medium near the pumping spot induce changes in the refractive index, leading to variations in linear polarization. Consequently, with increasing offsets, both the laser mode and its associated polarization state undergo corresponding transitions. The polarization states of different HG modes recorded by the POL and QWP is depicted in Fig. S1. Furthermore, it becomes apparent that when offsets exceed 100 µm, the laser mode ceases to maintain linear polarization. Instead, the polarization state becomes complex, displaying not only linear but also elliptical characteristics, as illustrated in Fig. S1(c).

 

Fig. 1. Experimental setup and polarization results of Hermite-Gaussian modes. (a) Schematic diagram of experimental setup. (b) presents HG modes and their corresponding polarization state. The arrows, represented by different colors, correspond to the color scale ranging from 48° to 72°. (c) and (d) depict the variations of polarization angle and. divergence angle with order of the HG mode, respectively.

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Considering the birefringence of the c-cut $\textrm{Nd}:\textrm{YV}{\textrm{O}_4}$ crystal, when a ray normally incident into the crystal, it experiences refractive index ${n_o}$ as denoted by the red line in Fig. 2(a). When the ray incident with an angle $\theta $ (green line), the ray splits into ${n_o}$ ray (orange line) and ${n_{eff}}$ ray (blue line) inside the crystal. The effective refractive index, ${n_{eff}}$, can be expressed as ${n_{eff}} = \frac{{{n_o}{n_e}}}{{\sqrt {n_e^2{{\cos }^2}\theta + n_o^2{{\sin }^2}\theta } }}$. The refractive angle, ${\theta _{eff}}$, changes in response to variations in ${n_{eff}}$ according to Snell's law, resulting in the transitions of the optical path within the crystal. The ${n_o}$ ray and ${n_{eff}}$ ray exhibit orthogonal polarization states and accumulate a phase difference while propagating through the crystal, resulting in a phase retardation $\mathrm{\delta }$ between the two orthogonal polarization states given by Eq. (1).

As a result, the polarization state inside the cavity with different trajectories can be different after the ray passing through the gain medium, as shown in Fig. 2(b). As an incident angle $\theta $ increase, the value of ${n_{eff}}$ gradually vary from 1.9573 (${n_o}$) toward 2.1652 (${n_e}$), as shown in Fig. 2(c). Figure 2(d) displays the phase retardation and the corresponding polarization states associated with different values of $\theta $ within the dashed square shown in Fig. 2(c). Consequently, the birefringence of the gain medium enables the tuning of the polarization sates of the laser modes through the adjustment of $\theta $. Conclusive evidence is apparent in Fig. S1(c), the center region of the mode possesses smaller divergence angle, resulting a linearly polarized. The outer regions of the mode exhibit larger divergence angles, causing a transition in the polarization states to elliptical polarization. Additionally, the raw data containing basic information for the HG modes can be found in Table S1.

 

Fig. 2. Polarization tuning by birefringence crystal. (a) demonstrates the ray propagate in the birefringence crystal with different refractive indexes. The polarization transition is depicted in (b). (c) and (d) describe the variation of ${n_{eff}}$ and phase retardation $\delta $ in relation to different incident angle $\theta $, respectively.

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Thus far, we have explored the generation of Hermite-Gaussian modes and observed transformations in both the modes and their corresponding polarization states through the use of offset pumping. However, the tunability of the HG mode is limited, emphasizing the need for further exploration of alternative modes to achieve polarization tuning.

At specific cavity lengths, the laser cavity generates a mode that is formed by the superposition of multiple HG modes with the same energy called degenerate mode, and the cavity in which it occurs is known as a degenerate cavity. The laser modes’ frequency is given by $f({n,m,l} )= \mathrm{\Delta }{f_L}[{l + ({m + n + 1} )\times ({\mathrm{\Delta }{f_T}\textrm{ / }\mathrm{\Delta }{f_L}} )} ]$, m and n are the transverse mode indices, l is the longitudinal mode index. $\mathrm{\Delta }{f_T}$ and $\mathrm{\Delta }{f_L}$ represent the transverse and longitudinal mode spacing, respectively, with $\mathrm{\Delta }{f_L}$ defined as $c\textrm{ / }2L$ [26,40]. In a plano-cylindrical cavity, the degenerate mode is represented by $\mathrm{\Omega }$ and can be mathematically described by Eq. (2).

The parameters L and R correspond to the cavity length and the radius of curvature of the cylindrical mirror, respectively. $P,Q$ are co-prime integer, when $\mathrm{\Omega }$ is a simplest rational number like $1/3,2/5$ etc. With the offset pumping at the degenerate cavity, the laser beam is different from the conventional high order HG beam. When the cavity length is set to degenerate cavity, the lasing mode become degenerate with offset pumping leading to geometric trajectory. Figure 3(a) and 3(b) display the experimental tomography conducted from $\textrm{z} = $-L to $\textrm{z} = $1.5 L for different degenerate states of $\mathrm{\Omega } = 1/3$ (Z mode) and $1/4\; $(W mode). Upon analyzing the geometric trajectory of the Z mode within the cavity, it was observed that there are three distinct beams propagating towards the z direction. These three beams include a red path which corresponds to a normally incident ray, and two green paths with identical incident angles on the gain medium. In the W mode shown in Fig. 3(b), four beams possess identical incident angles (green path) on the gain medium. Therefore, it can be concluded that coherent multi-beams can be achieved in the degenerate cavities. The dashed circles shown in Fig. 3(a) and 3(b) depict the area of the pump position, leading to localized variations in the thermal gradient, thereby triggering thermal effects and resulting in an asymmetry in polarization differences.

 

Fig. 3. Trajectory of degenerate states. (a) and (b) show the intensity distribution of Z mode and W mode at different cavity positions tracked by the reimaging lens. The cavity length L (mm) and offsets $\Delta y$ (mm) is set to be $({L,\; \Delta y} )= $ (7.5, 0.12) and (5, 0.17), respectively.

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By leveraging geometric configurations and birefringence, enhanced control over polarization can be achieved by progressively increasing the offsets and cavity lengths. The polarization states under different offsets and cavity lengths of Z mode and W mode are shown in Fig. 4(a) and 4(b), respectively. For the sake of clarity, all laser patterns have been rotated by 90 degrees in their presentation. As the offsets increase, the polarization state of the central spot of the Z mode (represented by the red path in Fig. 3(a)) remains linearly polarized. However, the other two spots’ polarization states transit from being identically linearly polarized to being elliptically polarized with opposite handedness when the incident angle is larger than $2.7^\circ $.

During a roundtrip, the light path experiences no phase retardation for normal incidence into the birefringence crystal. For the bottom and upper spots, the light passes through the crystal with birefringence $({4\textrm{n} + 1} )$ and $({4\textrm{n} + 3} )$ times, respectively, meaning the light path experiences $({2\textrm{n} + 1/2} )\pi $ and $(2\textrm{n} + 3/2)\pi $ phase retardations. As a result, a $\pi $ phase shift occurs between the two spots, leading to an opposite rotation of elliptical polarization. The elliptical polarization gradually changes as the offsets increase. Geometric beams, formed by superposed HG modes under larger off-axis conditions, have a larger divergence angle. Within the gain medium, the geometric beam separates into distinct regions, with only one beam influenced by the thermal effects of the pumping beam. In this condition, the birefringence effect dominates, resulting in elliptical polarization states. With sufficiently large offsets ($\mathrm{\theta } > $ 5.0$^\circ $), the beam becomes linearly polarized again. Furthermore, there is a $\pi $ phase shift in the polarization state for all beams when the cavity length is extended by µm. This is because altering the cavity length determines whether the cavity is in the ${n_o}$ or ${n_e}$ domain due to different stability criteria, resulting in the phase shift. For the case of W mode, the polarization results are recorded at $\textrm{z} = $1.5 L since the four laser beams combine to two beams at the far field. Similar to the Z mode, the W mode exhibits a transition in polarization states from linear to elliptical as the offset increases, and the polarization state transit with a $\mathrm{\pi }$ phase shift as the cavity length increase for 60 µm. Conspicuously, while all four beams exhibit an identical divergence angle, the polarization discrepancies of the beams originated from the pumping spot gradually manifest as the offset increases.

 

Fig. 4. Polarization results of the degenerate states. (a) and (b) show the far-field intensities and the polarization states of each ray for $\mathrm{\Omega } = 1/3$ and $1/4$ at different cavity lengths and offsets, respectively. The variation of cavity length is 60 µm for both the degenerate states.

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The mechanism for tuning the polarization states of the degenerate laser mode is straightforward based on the geometric optics and birefringence crystal, as presented in Fig. 2(b). By adjusting the incident angle of the gain medium and tuning the polarization states of the emitted beam, the phase retardation of the output beams can be managed in the birefringent laser system. This system generates multiple coherent beams and offers the ability to tune the polarization states. The detailed divergence angle and phase retardation values are provided in Table S2.

3. Polarization analyses and discussion

The polarization states are determined by the thermal effect and birefringent effect of the gain medium. In the case of HG mode with a small angle, the thermal effect dominates the polarization. However, the increasing offset (divergent angle) also involves the birefringent effect. Figure 2(d) depicts the relationship between the phase retardation δ and changes in divergence angles. Notably, the phase retardation does not exhibit linear variation. In different intervals of divergence angles, despite an equal increase in angle, there are variations in the change of phase difference. Hence, as the order increases, HG modes accumulate different phase retardation variations, leading to polarization states with the two slopes due to thermal effect and birefringent effect. In order to analyze the trend of the polarization state of the laser mode, the method below help us to realize the behavior of the polarization states. To demonstrate the evolution of the polarization states and facilitate numerical analysis, the polarization states are characterized using the long axis angle ${\theta _L}$ and ellipticity $\chi $, and map onto the Poincaré sphere, as depicted in Fig. 5(a), and 5(b). When $\chi $ reaches zero, the ellipse transitions into a line, indicating a linear polarization state with an angle of ${\theta _L}$. As the value of $\chi $ increases, the elliptical shape becomes more rounded, eventually transitioning into a circle at ${\pm} 45^\circ $. The positive and negative values of $\chi $ correspond to right-handed and left-handed polarization, respectively. Figure 5(b) displays the polarization states with various combinations of ${\theta _L}$ and $\chi $ representing different configurations. These two parameters are also associated with the longitude and latitude coordinates on the Poincaré sphere, respectively, as depicted in Fig. 5(b).

In Fig. 5(c) and 5(d), the polarization states for the Z mode and W mode from Fig. 4 are depicted on the Poincaré sphere, respectively. The coordinate of each states are shown in Table. S3 and Table. S4. The numbers marked in the inset of the laser modes indicate different optical paths. The Poincaré spheres depict the evolution of the polarization states for the two modes as the offsets and cavity length increase. The dots of varying colors on the Poincaré sphere correspond to different offsets, with a guide arrow providing a clear illustration of the evolution that takes place as the offsets increase. For both cavity lengths, the Z mode of the normally incident ray (path 2) is aligned with the equator, while the ray with a divergence angle (path 1, 3) becomes increasingly elliptical for polarization states on different hemispheres (helicities) but with the same trend. The polarization outcomes for the long cavity length display a similar trend but with a phase difference of $\pi $ compared to the short cavity length. When it comes to the W mode, path 1 and path 3 have comparable trajectories, whereas path 2 and path 4 share a resemblance for both cavity lengths. Although all four paths are anticipated to have the same phase retardation and polarization state since they have the same divergence angle, a thermal impact on the pumping spot leads to a variation in the refractive index, which results in distinct phase retardation and polarization outcomes. A similar phenomenon can be observed in $\mathrm{\Omega } = 1/5$, as depicted in Fig. S2.

 

Fig. 5. Polarization states on Poincaré sphere. (a) shows the definition of long axis ${\theta _L}$ and ellipticity $\chi $. Polarization with different parameters are mapped onto the Poincaré sphere, as depicted in (b). (c) and (d) presents the polarization states of Z modes and W modes mapped onto the Poincaré sphere, respectively. The dots with colors on Poincaré sphere show the variation of the polarization states in Fig. 4, dark arrows with red numbers represent different polarization evolution of multiple optical beams as offsets increase.

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Through a systematic adjustment of offsets and cavity length, stable and replicable lasing of geometric modes exhibiting diverse polarization states is achieved. This holds significant implications for applications in communications and security encryption. An external mode converter, located beyond a cavity can facilitate conversion of HG modes into Laguerre-Gaussian (LG) modes, and transfigures geometric modes into a superposition of LG modes [41,42]. This transformation imbues the modes with optical orbital angular momentum (OAM). In this study, we investigate the generation and manipulation of various of polarization states, which can potentially be combined with OAM to introduce a new degree of freedom for applications in the interactions of light and matter, as well as optical tweezing capabilities. Distinct helicities for polarization outcomes are manifest on the opposite hemisphere of the Poincaré sphere, and these can be mapped onto a Bloch sphere in an analogous fashion. Consequently, this meticulously designed setup holds the potential to generate well-manageable optical polarization states, offering a spectrum of applications.

4. Conclusion

In conclusion, this study presents the outcomes of the precise generation and modulation of laser polarization states within a simplified cylindrical laser cavity, achieved through the integration of a birefringent crystal. The results illuminated the possibility of modulating various polarization states by manipulating pumping offsets and cavity lengths, accompanied by a numerical analysis based on phase retardation. Increasing the pumping offset effectively governs the linear polarization direction of higher-order Hermite-Gaussian modes. Moreover, precise manipulation of the cavity length enables the rigorous control of multiple polarization states in geometric beams, transitioning from linear polarization to elliptical polarization. The analysis method of mapping laser polarization states onto the Poincaré sphere provides a more comprehensive understanding of the evolution of polarization states. This study provides valuable insights and prospects regarding the utilization of polarization degrees of freedom in optical design and quantum optical applications.