Lasers with structured output light are characterized by compact structure, high robustness, power scaling capability, and improved beam quality^[1]. The ability to tailor the properties of light such as intensity, phase, and orbital angular momentum (OAM) directly from the laser source has paved the way for frontier technologies, including particle manipulation^[2,3], high-security encryption^[4], quantum entanglement, and communication^[5,6]. Previous endeavors in designing structured-light lasers fall into two broad classifications. The first category involves manipulating the gain-to-mode overlap through the shaping of pump light beams^[7–10], off-axis, or tilted pumping^[11–13], thereby favoring specific modes in mode competition and enabling selective output of structured light from lasers. The second category leverages diverse mode selection mechanisms. These include amplitude-based components like spot-defect mirrors^[14,15] or amplitude masks^[16,17], as well as phase-based elements such as diffractive optical elements^[18–20], axicon prisms^[21], short-focal lenses^[22], and diffraction grating^[23]. Specialized devices, including astigmatic mode converters^[24], wave plates and q plates^[25–27], and interference output couplers^[28,29] also find application. Moreover, the use of metasurfaces^[30,31] and spatial light modulators (SLMs)^[32,33] is dynamically reshaping the landscape of structured-light laser design.

In practice, the above methods with static configurations can only produce a specific type of field. While certain tunable laser setups can generate either a family of Hermite–Gaussian (HG) modes^[11,12] or a set of Laguerre–Gaussian (LG) modes^[16,17], they generally fall short in creating arbitrary complex fields. In 2013, the structure of the “digital laser” was proposed, which could generate many kinds of fields^[32]. However, due to its limited mode-discrimination ability, this cavity faces challenges in generating some significant fields. For example, it cannot control the handedness of LG modes, which is significant in applications such as particle manipulation and optical communication. The main reason is that, in the digital laser, the mode discrimination is realized by manipulation of intracavity losses, whereas LG modes with opposite OAMs have identical transverse intensity distribution, and suffer the same intra-cavity losses. As a result, it is hard to distinguish the handedness of LG modes in such a laser configuration.

In this work, we provide an alternative laser configuration with robust mode-discrimination ability, which allows for the generation of arbitrary complex fields directly from the cavity. Compared to extracavity approaches, the intracavity method proposed herein for generating structured-light fields boasts unique advantages of compactness and flexibility. It eliminates the need for additional beam shaping or spatial filtering structures and avoids the requirement for re-calibration, while still enabling the production of a variety of distinct light-field structures. The essence is to enforce a fundamental Gaussian mode to oscillate and self-reproduce, while the desired field is separated from the self-reproduction mode and generated at the first diffraction order. This makes it convenient for manipulating the amplitude and phase of the output field. We verify the proposed method by constructing an Nd:YAG ring laser cavity implementing an SLM as the cascaded modulation device. A series of typical HG modes, LG modes, flat-top mode, and amplitude-only pattern “A” can be generated from the same ring laser configuration. Finally, we perform an interferometric measurement on the topological charges of the output vortex LG modes, further validating the fully selectable control of handedness by our method.

In Fig. 1(a), we illustrate the basic idea of the ring laser resonator based on the cascaded phase modulation scheme. The SLM and a high-reflective mirror (M3) constitute the cascaded modulation configuration. Figures 1(b) and 1(c) schematically demonstrate the two different-order beam propagation processes within the ring laser configuration. First, the zeroth-order beam f0 is the self-reproduction field as depicted in Fig. 1(b). The self-reproduction field f0 is modulated sequentially by two cascaded phase holograms, ϕ1 and ϕ2. Then the modulated field f2 is Fourier-transformed by a lens. The field distribution of the zeroth-order beam in the Fourier plane is identical to that of the initial field f0, satisfying the self-reproducing relationship. The presence of cascaded phases establishes a natural mechanism for unidirectional operation. Only the field f0 can achieve self-reproduction within a single transit through the cavity. Other fields, or those propagating in the opposite direction, undergo rapid divergence upon experiencing the cascaded phase modulation, thereby precluding the formation of laser oscillation. Second, the desired output field fout is the first-order beam in the Fourier plane as shown in Fig. 1(c).

We employ a quasi-Newton gradient-descent algorithm to obtain the cascaded modulation phase ϕ1 and ϕ2, which control the beam propagation in the laser cavity. Figure 2 schematically illustrates the calculation model. The initial Gaussian field f0(x0,y0) propagates for a distance L1 to f1(x1,y1). This field subsequently encounters cascaded phase holograms, ϕ1 and ϕ2, spaced by L2, which impart phase functions of exp[iϕ1(x1,y1)] and exp[iϕ2(x2,y2)], respectively. The modulated field, f2(x2,y2)exp[iϕ2(x2,y2)], is then Fourier-transformed to F(xF,yF) by the lens. Here, notation (xi,yi) with i=0,1,2,F denotes transverse coordinates in respective planes. Under the paraxial approximation, the beam propagation can be expressed as {f1=∬f0iλL1exp{iπ[(x1−x0)2+(y1−y0)2]λL1}dx0dy0f2=∬f1exp(iϕ1)iλL2exp{iπ[(x2−x1)2+(y2−y1)2]λL2}dx1dy1F=∬f2exp(iϕ2)iλfexp[2iπ(xFx2+yFy2)λf]dx2dy2.(1)

The transformed field F(xF,yF) in the Fourier plane is mainly composed of two diffraction orders. The zeroth-order beam F0 is designed to be identical to the initial field f0. The first-order beam F1 is the desired output field with controllable amplitude and phase. The cost function of the optimization algorithm is defined as the difference between the transformed fields and the target fields as {L(ϕ)=L1(ϕ)+L2(ϕ)L1(ϕ)=‖F0−α1f0‖22L2(ϕ)=‖F1−α2fT‖22,(2)where L(ϕ) denotes the overall cost function of the optimization algorithm, with L1(ϕ) and L2(ϕ) representing the cost functions associated with the intracavity and extracavity mode conversion processes, respectively. The symbol ‖.‖22 signifies the F-norm calculation. ϕ encompasses the phase holograms ϕ1 and ϕ2. The constants α1 and α2 are positive coefficients related to power efficiency, specifically as α12 and α22. fT symbolizes the target output field. The objective of the optimization algorithm is to find the cascaded phase hologram ϕ that minimizes L(ϕ). Cascaded phase modulation has been proven to facilitate highly efficient and precise optical field modulation^[34]. Therefore, through manipulation of ϕ1 and ϕ2, it is possible to closely match the amplitude and phase distribution of F0 with f0 while maintaining a high mode conversion efficiency. This approach permits the generation of a self-reproduction field within the laser cavity while also enabling arbitrary control over the amplitude and phase of the output field F1. Comprehensive details regarding the optimization algorithm are presented in Supplement 1.

The experimental setup for the ring laser resonator employing a cascaded modulation scheme is illustrated in Fig. 3. The gain medium is a 0.8% doped Nd:YAG crystal rod with a dimension of 67 mm (length) by 4 mm (diameter). The SLM, Hamamatsu LCOS X15213-03 (1272 pixel × 1024 pixel, 12.5 µm pixel size, and 97% light utilization efficiency at 1064 nm), is tilted at an angle of 3°. The focal length of the lens is 200 mm. The total length of the cavity is 1005 mm and converted to an effective length of 975 mm due to the refractive index of the gain medium. The effective lengths of L1 and L2, as depicted in Fig. 2, are 423 and 152 mm, respectively.

Figure 4 graphically illustrates the experimental results of the on-demand generation of various fields directly from the ring laser cavity: a typical HG22 mode, an LG08* mode, a flat-top mode, and an amplitude-only pattern “A” directly from the ring laser cavity. The superscript (·)* denotes the petal-like beam from the superposition of LGp,+l and LGp,−l modes. The intensity of the output fields generated from the laser are 1:1 imaged by a 4f system (not shown) to a CCD camera (Spiricon SP928). The cascaded phase holograms for generating such different kinds of modes are shown in the first column of Fig. 4, while their corresponding two-dimensional intensity profiles are exhibited in the second column. To better illustrate the performance of the cascaded modulation cavity configuration, the axial intensity distribution of each output mode is compared with that of the target mode predefined in the optimization algorithm. As evidenced in the third column of Fig. 4, the close agreement between the output fields and their intended targets confirms the ring laser cavity’s capability of producing complex fields with controllable amplitude and phase.

In addition to the above four different fields, a series of HG and LG modes generated directly from the ring laser cavity is presented in Fig. 5, which further demonstrates the capability of the ring laser configuration to generate arbitrary complex fields. Each HG and LG mode features an output power within the range of 100 mW and a fidelity surpassing 90%. The observed fidelity is slightly lower than simulated results, primarily due to constraints imposed by the dimensions of the Nd:YAG rod medium in our designed laser. This limitation restricts the usable area of the SLM, preventing full exploitation of its potential (hence the slight discrepancy in achieved fidelity).

To further highlight the advancements of our proposed cascaded modulation ring laser cavity scheme, we generate LG modes with opposing handedness directly from the cavity and proceed to conduct interferometric experiments to quantify their topological charge numbers. Figure 6(a) illustrates the setup for this interference experiment. The output LG mode is first split into two beams by beam splitter 1 (BS1). One of them is directly transmitted to beam splitter 2 (BS2) and captured by a CCD camera. The other is filtered by an aperture (AP) and expanded by a telescope configuration comprised of lens1 (50 mm focal length) and lens2 (200 mm focal length). The to-be-measured LG mode and the reference beam are rejoined at BS2 at a small tilted angle. The interference fringes are recorded by a CCD camera. For illustrative purposes, we present the results for LG0,+1 and LG0,−1 modes. The cascaded phase holograms for the generation of the two modes with opposite topological charges are depicted in Figs. 6(b) and 6(e). The output doughnut-like intensity profiles are displayed in Figs. 6(c) and 6(f). Notably, the interference patterns in Figs. 6(d) and 6(g) exhibit clear fork-like fringes orientated in opposing directions, affirming the modes’ differing topological charges. The difference in the number of forked stripes is 1, indicating the topological charge of |l|=1. Thus, the interference experiment confirmed that the structured light with controllable topological charge can be directly generated from the ring laser cavity using a cascaded modulation scheme.

In conclusion, we propose what we believe is a novel laser configuration based on a cascaded modulation scheme, which allows for the on-demand generation of arbitrary complex fields directly from the laser. By designing the cascaded phase holograms, the fundamental Gaussian mode is capable of oscillating in a round trip, whereas the desired mode is spatially separated and outputs from the laser. In the verification experiment, a variety of typical HG modes, LG modes, a flat-top mode, and an amplitude-only pattern “A” are directly generated from the laser. Furthermore, we perform the interference experiment to measure the topological charges of two LG modes carrying opposite OAMs. The experimental results have confirmed that the laser cavity via a cascaded modulation scheme is able to generate arbitrary complex fields, including a fully selectable control of handedness of vortex LG fields. In this ring laser structure, the incorporation of additional cascaded modulation layers holds the potential to achieve higher precision and efficiency in generating complex fields or exerting control over beams in more diffraction orders. We believe this versatile laser configuration may lead to new and interesting applications for its ability to tailor the amplitude, phase, and OAM of light directly from a laser source.