Structured light has received ever-increasing attention in recent years especially for the direction to extend more degrees of freedom and higher dimensions in optical vortices carrying orbital angular momentum (OAM) [1–3]. Conventional structured light modes were usually studied as two-dimensional (2D) transverse patterns as eigenmodes of the paraxial wave equation, which typically include the Hermite–Gaussian (HG) modes with two indices of $x$- and $y$-coordinates, and Laguerre–Gaussian (LG) modes with radial and azimuthal (or OAM) indices as typical cases of vortex beams [4]. Recently, many novel forms of optical vortices were created to revolutionize related applications. For instance, the polygonal-shaped vortex beams with additional symmetry control rather than the circular vortex beams were designed to achieve diversified microstructure fabrication and manipulation [5–8]. Beyond the prior single-singularity vortex beams, the multi-singularity geometric vortices were created and used to extend larger capacity in quantum and classical information transfers [9–13].

The diversified 2D higher-order eigenmodes, such as HG, LG, and tunable Hermite–Laguerre–Gaussian (HLG) modes, were already widely studied and generated with control from customized compact laser sources [14–18]. The creation of novel three-dimensional (3D) structured light and the controlled generation from lasers are still highly desired. As a typical 3D structured light, the geometric modes with salient properties such as quantum–analog coherent states and ray–wave duality have broadened the horizon in manipulating exotic symmetry and structures of light with multiple degrees of freedom [9,19–22]. Such geometric modes can be directly generated from a simple plane–concave cavity, and diversified geometric patterns and trajectories can be tuned by controlling cavity parameters and frequency degeneracy [23–25]. As a typical example, the multiaxial geometric (MAG) modes can enable optical vortices to be coupled on multiaxial geometric rays, forming exotic multiaxial geometric vortices with tunability of both central OAM and sub-OAM on geometric trajectories beyond the limit of conventional OAM beams [26–29], which were already used in applications of optical communications toward larger information capacity [10–13].

In this Letter, we further break the limitation of the tunability of MAG modes, and in particular, we achieve the control of both radial and OAM indices on each axis, termed multiaxial super-geometric (MASG) modes. We also present the laser setup for controlled generation of MASG modes with tunable topologies of HLG bases, which is achieved by realizing superposed SU(2) coherent state modes with opposite trajectory phases exploiting selective pumping and intracavity mode conversion. We also studied the propagation of MASG laser beams, especially the LG-based case with complex OAM coupled by multiaxial geometry. Our MASG laser opens new dimensions for structured light control with great potential for advanced applications such as optical trapping, manufacturing, and communications.

The MAG modes fulfill the form of SU(2) coherent states, which can be expressed as specific superposed wave packets of a class of eigenmodes of a laser resonator [21]

However, it is only possible to control a single OAM index without chirality control for the sub-axial mode. To overcome this limit, we design the MASG modes, where we can control arbitrary sub-OAM and sub-radial indices coupled to multiaxial geometry, which can be obtained as

 

Fig. 1. (a) A MASG mode is obtained by specific superposition of SU(2) geometric modes. The left and right columns correspond to the HG-based and LG-based modes, respectively. (b) The HG-based (left) super-geometric modes with different values of index $n_{0}$ and different values of $J$ of superposed SU(2) geometric modes can be converted to LG-based (right) MASG vortex modes carrying various sub-radial and sub-OAM topological orders, $p_s$ and $\ell _s$, on each geometric axis.

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The experimental setup of the MASG laser is shown as Fig. 2(a) in the gray frame. A pair of cylindrical lenses is set into a concave–concave cavity to break the symmetry of the cavity so that the indices-controlled 2D modes can be generated from the cavity [15], in addition to a Yb:CALGO crystal as the gain crystal. The 2D modes can be tuned in the $x$-direction and the $y$-direction independently by controlling the off-axis displacements of the first cylindrical lens in the $x'$-direction and R1 in the $y'$-direction, respectively. The focal length of the cylindrical lenses in the $x$-direction is infinity, and that in the $y$-direction is $f$, which correspond to the structures of the cavity in the $x$- and $y$-directions as shown in Figs. 2(b) and 2(c). With delicate controls of the distances $d_{1}$, $d_{2}$, and $d_{3}$ in Fig. 2(c), the laser cavity can fulfill the frequency-degenerate condition and emit the geometric modes as SU(2) coherent state [21], as well as their superposed modes formed by the geometric modes with different coherent-state phases, so that MASG modes can be generated. The generated HG-based MASG modes from the cavity are split from the light path by BS at position 1 in Fig. 2(a), and the transmitted beam is converted into LG-based MASG modes by an extracavity mode converter as position 2.

 

Fig. 2. (a) Experimental setup: a laser is formed by two cavity mirrors R1 [AR at 979 nm and high-reflective (HR) at 1040–1080 nm, radius of curvature $R_{1}$: 1200 mm] and R2 [output coupler (radius of curvature $R_{2}$: 200 mm)], a gain crystal (Yb:CALGO) and two intracavity cylindrical lenses (focal length $f=25$ mm). The $x$-axis is parallel to the generatrices of cylindrical lenses, and $\Delta {d}$ represents the off-axis displacement of R1 along the $y'$-axis ($45^{\circ }$ difference from the $y$-axis). A dichroic mirror (99% reflectance at 979 nm and 90% transmittance at 1050–1080 nm) is used to block the pump light. A beam splitter (BS) splits the beam into two paths, one for collecting HG-based modes (position 1). For another path (position 2), an extracavity mode conversion including three lenses L1 to L3 (focal lengths: 30 mm, 100 mm, and 100 mm) and two cylindrical lenses ($f=25$ mm) with generatrices parallel to the $y'$-axis. (b) The optical path in the cavity along the $x$-axis includes $R_{1}$, $R_{2}$, and cavity length $L$, while, (c), that along the $y$-axis includes focusing lenses ($f$) and distances between cavity elements ($d_{1}$ to $d_{3}$).

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The pattern evolution of an SU(2) geometric mode in cavity is affected by the initial phase $\phi _{0}$ which has a causal relationship with the localization on the geometrical trajectories. The initial phase is decided by the matching between trajectory and gain in cavity, related to the position of the gain crystal. In our experiment, we control the cavity geometry into the case of $(q,p)=(4,0)$, so that frequency-degenerate eigenmodes having different $m$ indices as well as the same $n$ indices with the trajectory evolution in the $x$-direction can be excited in the cavity. Different trajectories with different initial phases can be obtained with the ABCD matrix $\textbf {X}$ of the cavity in the $x$-direction as Fig. 3(a). The $x$-axis in Fig. 3(a) is the position of the waist of the beam in cavity, and $d_{0}=z_{R}\tan (\phi _{0}/4)$ represents the position with $(x_{0},\theta _{0})=(0,\lambda \sqrt {M+1}/\pi /\omega _{0})$ [23], where $\theta$ is the angle of trajectory with the propagation direction. Therefore, the whole trajectories can be obtained as $(x,\theta )^{\text {T}}=\textbf {X} (x_{0},\theta _{0})^{\text {T}}$. As the overlapping drawing of these two trajectories at the bottom of Fig. 3(a), the coincidence black point is the position of the gain crystal where these two trajectories have the same matching to gain in cavity and are excited simultaneously, resulting in the generation of MASG modes. The corresponding coupled wave packet patterns in cavity are shown in Fig. 3(b), respectively, where the wave packet of the MASG modes (at the bottom) matches well with the periodic trajectory.

 

Fig. 3. (a) In a frequency-degenerate laser cavity, SU(2) geometric modes can have different ballistic trajectory characteristics with different initial phases (e.g., the blue and red) and their superposed states, (b) which correspond to different geometric wave packet distributions in cavity. As the superposition of two geometric modes in the frame, a MASG mode can be formed as below, and the laser gain position should be set on the coincident position (black dot) of blue and red lines to stimulate two modes simultaneously in the cavity.

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The simulated and experimental results of MASG modes are shown in Fig. 4, with the hot and gray colorbars, respectively. With the superposition of two SU(2) coherent state modes with $\pi /2$ initial phase difference, the MASG modes with the indices $p_{s}=0$, $\ell _{s}=-1$ in the $y$-direction are shown in Fig. 4(a). The HG-based modes are in the upper row of Fig. 4(a) which are collected from the split beam in position 1 of Fig. 2(a) and the LG-based modes in the second row are collected in position 2. With a larger off-axis displacement of R1 ($\Delta {d}=1.641$ mm) based on the cavity structure of the MASG modes with initial indices $p_{s}=0$, $\ell _{s}=-1$, the MASG modes with $p_{s}=1$, $\ell _{s}=0$ can be generated, as shown in Fig. 4(b). The tunability of the radial parameter of each LG-like distribution on each axis of the MASG modes is achieved, and the simulated results and experimental results match well. The interference patterns of the MASG modes are also collected to showcase the phase and singularity distributions of the MASG modes with $p_{s}=1$, $\ell _{s}=0$ as shown in Fig. 4(c).

 

Fig. 4. The simulated results (the hot colorbar) and experimental results (the gray colorbar) of the HG-based (the upper) and the LG-based (the lower) MASG modes with (a) indices $p_{s}=0$, $\ell _{s}=-1$ and (b) $p_{s}=1$, $\ell _{s}=0$ for different off-axis displacements of R1. (c) The simulated and experimental results of the interference patterns. The inserts show an enlarged view of the coherent fringes.

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The simulated propagation of the LG-based MASG modes with $p_{s}=1$, $\ell _{s}=0$ is shown in Fig. 5(a), revealing its 3D multiaxial geometric trajectory and stability of the wave packet. The selective results of transverse pattern and phase at $x$–$y$ cross sections at $z=0$, $z=1.8z_{R}$, and $z=3z_{R}$ are shown in Fig. 5(b). The experimental results are collected after the extracavity mode conversion cavity, which agree with the simulated results with strong central OAM effect. In other words, the large topological charge of the central vortex phase singularity induces the twisting wavefront to twist multi-ray geometry upon propagation.

 

Fig. 5. (a) The simulated 3D twisted evolution of LG-based MASG modes shows the trajectory of each geometric axis. (b) The intensity distributions of theoretical and experimental intensity patterns and phase at selective cross sections upon propagation of $z=0$, $z=1.8z_{R}$, and $z=3z_{R}$.

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In addition, with controlled rotations of the extracavity cylindrical lenses, the arbitrary HLG-based MASG modes can be controlled from the HG-based MASG modes with different mode astigmatism parameters $(\alpha,\beta )$. Figure 6 represents the simulated and experimental results of the intensity distributions, together with the predicted phase distributions, of the astigmatic HLG-based MASG modes with $p_{s}=1$, $\ell _{s}=0$, revealing their ability of complex OAM control.

 

Fig. 6. The evolution of MASG modes versus different degrees of mode conversion via the rotation of two extracavity cylindrical lenses (theoretical and experimental intensity patterns and phase), under the control of different astigmatic parameters $\alpha$ and $\beta$.

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In conclusion, we present the creation and control of a new class of 3D structured light, i.e., MASG modes, from a compact laser. The MASG is theoretically achieved by a superposition of multiple SU(2) coherent states of eigenmodes with different initial phases in the cavity matching the trajectories and gain relationships. The MASG modes are tunable in both radial and OAM indices of LG-like distribution on each axis, which break the limit of the previous MAG vortex modes, and open up new degrees of freedom in structured light control. The tunability of multiple topological orders coupled with geometric trajectory will hatch new applications of multi-body classical entanglement, new encoding protocols for large-capacity communication, and new dimensions for optical manipulations in light–matter interaction.