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Abstract
Review Article Evolution on spatial patterns of structured laser beams: from spontaneous organization to multiple transformations a,b,c,† a,b,c, ,† d,e f a,b,c a,b,c a,b,c a,b,c Xin Wang, Zilong Zhang, * Xing Fu, Adnan Khan, Suyi Zhao, Yuan Gao, Yuchen Jie , Wei He, a,b,c d,e, a,b,c Xiaotian Li, Qiang Liu, * and Changming Zhao Beijing Institute of Technology, School of Optics and Photonics, Beijing, China Ministry of Education, Key Laboratory of Photoelectronic Imaging Technology and System, Beijing, China Ministry of Industry and Information Technology, Key Laboratory of Photonics Information Technology, Beijing, China Tsinghua University, Ministry of Education, Key Laboratory of Photonic Control Technology, Beijing, China Tsinghua University, State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Beijing, China Nankai University, School of Physics, Key Laboratory of Weak Light Nonlinear Photonics, Tianjin, China Abstract. Spatial patterns are a significant characteristic of lasers. The knowledge of spatial patterns of structured laser beams is rapidly expanding, along with the progress of studies on laser physics and tech- nology. Particularly in the last decades, owing to the in-depth attention on structured light with multiple degrees of freedom, the research on spatial and spatiotemporal structures of laser beams has been promptly developed. Such beams have hatched various breakthroughs in many fields, including imaging, microscopy, metrology, communication, optical trapping, and quantum information processing. Here, we would like to provide an overview of the extensive research on several areas relevant to spatial patterns of structured laser beams, from spontaneous organization to multiple transformations. These include the early theory of beam pattern formation based on the Maxwell–Bloch equations, the recent eigenmodes superposition theory based on the time-averaged Helmholtz equations, the beam patterns extension of ultrafast lasers to the spatio- temporal beam structures, and the structural transformations in the nonlinear frequency conversion process of structured beams. Keywords: spatial patterns; transverse modes; spatiotemporal beams; structured laser beams; nonlinear optics. Received Oct. 11, 2022; accepted for publication Jan. 6, 2023; published online Feb. 6, 2023. © The Authors. Published by SPIE and CLP under a Creative Commons Attribution 4.0 International License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI. [DOI: 10.1117/1.APN.2.2.024001] less emphasized. In the last 10 to 20 years, as a result of the 1 Introduction emergence of research on spatial characteristics of laser beams, Since their introduction 60 years ago, lasers with various char- especially, the orbital angular momentum (OAM), much more acteristics have developed rapidly, making them important light attention has been paid to structured laser beams with distinct 2–4 sources in various fields ranging from scientific research spatial or spatiotemporal structures. Such beams have brought 5–8 to industrial production. Almost all the characteristics of 24,25 various breakthroughs to many fields, including imaging, 9–14 15–18 a laser can be classified as temporal, spatial, or spectral 26,27 28–30 31–33 microscopy, metrology, communication, optical trap- 19–23 domain. Over the past few decades, much interest has been 34–36 37–39 ping, and quantum information processing. In the most given to laser properties in the temporal and spectral domains, recent five years, the number of reviews on structured light has while the spatial properties of laser beams seem to be relatively boomed, with the majority of reviews focusing on technical- 40–54 level research into the phenomenon, such as the generation 44,45 and detection technology of structured light, application in *Address all correspondence to Zilong Zhang, zlzhang@bit.edu.cn; Qiang Liu, 46,47 48 qiangliu@tsinghua.edu.cn the field of optical trapping and anti-turbulence, progress 49–51 52 These authors contributed equally to this work. in optical vortices and higher-dimensional structured light, Advanced Photonics Nexus 024001-1 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… as well as research on structured light in flat optics and non- with which they oscillate in a synchronized way. The locking linear optics. In contrast, there have been few studies on the concerns also the relative phases of the modes, so that the output evolution of the investigations and corresponding understand- intensity has a stationary transverse configuration. The patterns ings of the pattern formation of structured laser beams. Here, formed by the TML effect could possess phase singularities in we would like to take the spatial patterns as the main core to dark points. The theory of eigenmode superposition can also review the evolution and recent advancement on spatial patterns explain the formation of high-order complex transverse modes 87–89 of structured laser beams, from the early spontaneous organiza- and optical vortex lattices (OVLs). tion with numerical solutions of mathematical equations and For the second period of on-demand transformation of laser eigenmode superposition theories to the multiple transforma- patterns, a number of techniques have been developed in the tions of the spatiotemporal dimensions and nonlinear process of past 20 years to actively control the generation and transforma- structured laser beams. tion of laser beam patterns, giving rise to a better understanding The research on spatial patterns of structured laser beams of the spatial features of lasers. Numerous review articles on went through two periods: the first was the spontaneous organi- the actively controlled generation of structured laser patterns, 40,41 zation of patterns described by relative equations, while the using both intracavity oscillation and extracavity spatial 42–45 second was the transformation of laser patterns on demand. modulation methods, are readily available. The studies on Although there is no distinct separation between these two peri- the transformations of structured laser beams are mostly under ods, it is noticeable that over the past 10 years, we have steadily the premise of single longitudinal mode. However, if multiple gained a better understanding of how diverse laser spatial pat- longitudinal modes are involved, the time dimension is sup- terns originate and developed several effective techniques for posed to be reconsidered. Recently, research on spatiotemporal 103–114 producing spatial patterns on demand. Research on the forma- beams has sprung up. The direct generation of spatiotem- tion of structured laser patterns was a main focus in physics poral beams involves the principle of spatiotemporal mode lock- 55–69 from the 1960s to 1990s. The earliest research at that time ing, i.e., locking the laser longitudinal and transverse modes 69 70 combined Maxwell’s equations with Schrödinger’s equation, at the same time to form ultrashort pulses with special spatial leading to the laser amplitude E coupled with the collective intensity distribution. Spatiotemporal mode locking is often variables P and D for the atomic polarization and population obtained in fiber lasers, which realizes TML with the help of 71 107,115 inversion to describe the transverse mode formation. This spatial filtering, and longitudinal mode locking with the 62,72 116,117 set of equations is called the MB equations. Then, on the help of normal-dispersion mode-locking principle and a basis of the MB equations, which are a set of spatiotemporal saturable absorber. The spatiotemporal mode-locking beam multivariate nonlinear partial differential equations, the pattern opens up a new direction for the propagation and application formation characteristics of class A, B, and C lasers were of nonlinear waves. However, the spatiotemporal mode-locking 68,71 successively studied. The relevant equations are then further beams produced directly by fiber lasers often have irregular developed, and equations, such as the complex Ginzburg– intensity distributions. To obtain regular and more complex 73,74 75,76 Landau (CGL), complex Swift–Hohenberg (CSH), and spatiotemporal beams, a pulse shaper based on spatial light 77–80 Kuramoto–Sivashinsky (KS) equations were further derived. modulator (SLM) is applied, which can generate specific spa- 118–120 121 Through the numerical solution of these equations, the forma- tiotemporal optical vortices, spatiotemporal Airy beams, 122 123–126 tion of laser transverse patterns under specific parameters can be spatiotemporal Bessel beams, and so on. Another analyzed. In addition, with time and space terms involved, these method for actively controlling the generation and transforma- equations can explain both the spatial and temporal character- tion of structured laser beams is using nonlinear processes. istics of the patterns in some cases, including stability, oscilla- Combining nonlinear frequency conversion with the genera- 81–83 tion, chaos, and so on. However, most of these patterns are tion of structured laser beams, the beam patterns of harmonic ideal cases obtained under the condition of a single transverse waves are found to be endowed with much richer spatial in- mode of the laser. If multiple transverse modes with different formation. Through nonlinear frequency conversion of struc- frequencies are involved, it would be hard to analyze the tured laser beams, the beam pattern transformations in sum 127–130 results of multifrequency interaction through these equations. frequency generation (SFG), second-harmonic generation 131–140 141,142 Therefore, in the study of pattern formation in the last 20 years, (SHG), four-wave mixing (FWM), and other fre- 143–145 the analysis is often carried out through the applications of a quency upconversion processes are carried out. Usually, 84–89 set of eigenmode superposition theories. This set of theories these studies first generate structured beams with the help of mainly studies the spatial structure characteristics of the pat- modulation devices (such as SLM), and then carry out nonlinear terns. The basis of eigenmode superposition theory is the fun- conversion, which belongs to the external cavity nonlinear pro- damental composed modes, such as Hermite–Gaussian (HG), cess of structured laser beams. Meanwhile, for the intracavity Laguerre–Gaussian (LG), and Ince–Gaussian (IG) modes, nonlinear process of structured laser beams, some studies also 90–96 obtained by solving the Helmholtz equations, except with showed similar properties, while more complex and diverse 146–150 space terms and without time terms. Then, according to the spe- beam patterns can be obtained. cific laser cavity conditions and field distributions of the output The timeline of the evolution on spatial patterns of structured pattern, it could be analyzed whether coherent or incoherent laser beams is shown in Fig. 1. In this review, we highlight the superposition of the fundamental modes is generated through developments in laser spatial pattern studies with the emphasis the eigenmode superposition theory. This involves the concept on the spontaneous organization of pattern formation and the 97–99 of transverse mode locking (TML), namely, to lock the multiple transformations of laser spatial patterns. The original 100,101 phase or frequency of several transverse modes to obtain descriptive equations in the field of pattern formation are intro- the output beam. The cooperatively frequency-locked multi- duced in Sec. 2, including MB, CGL, CSH, and KS equations, mode regime, in which at least two transverse modes contribute which are fundamental to understanding the dynamics of pattern significantly to the output field, lock to a common frequency formation. Then, the current theories in the past 20 years are Advanced Photonics Nexus 024001-2 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 1 Timeline of the evolution on spatial patterns of structured laser beams. further discussed in Sec. 3, namely, the eigenmode superposi- 2 MB Equations-Based Pattern Formation tion theories. The superposition of transverse modes brings Pattern formation is a ubiquitous phenomenon in nature and a new vision in understanding laser physics and recognizing a phenomenon often found in laboratories; it was regarded as the vast possibilities for structured laser beam patterns. Results the spontaneous appearance of spatial order. Generally, all are analyzed and compared, covering coherent superpositions the patterns have something in common: they appear in spatially and incoherent superpositions. Then, in Sec. 4, more potential expanding dissipative systems, which are far from equilibrium developments are forecast in spatiotemporal laser beams, par- because of some external pressure. In optical systems, the ticularly in spatiotemporal mode locking in fiber lasers and mechanism of pattern formation is the interaction among spatiotemporal beams generated through pulse shapers based diffraction, partial resonance excitation, and nonlinearity. on SLM. In Sec. 5, various nonlinear processes of structured Diffraction is responsible for spatial coupling, which is neces- laser beams from external–cavity modulations to intracavity transformation are comprehensively reviewed. Finally, conclud- sary for the existence of nonuniform distribution of light fields. ing remarks and prospects are provided in Sec. 6. The role of nonlinearity is to select a specific pattern from Advanced Photonics Nexus 024001-3 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… several possible patterns. After reducing a specific model which can describe both pattern formation and time-domain into a simpler model, a common theoretical model describing properties. pattern formation is found to be the order parametric equation For class A lasers (e.g., He–Ne, Ar, Kr, and dye): 150,151 (OPE). Then, in the study of pattern formation in structured γ ≈ γ ≫ κ, the polarization and the population inversion ⊥ ∏ laser systems, the problem was precisely addressed through of the atom can be adiabatically eliminated; hence the system the description of optical resonators by OPE, which reflects the is only described by one field equation. The equation can be general characteristics of laser transverse patterns. simplified to the CGL equation, which is also the governing The exploration of the OPE of laser spatial pattern formation equation in superconductors and superfluids. Therefore, the last 55–57 can be traced back to the 1970s. By simplifying the laser two equations of Eq. (1) are eliminated, leaving only the non- equations for the class A case to the CGL equation, the relation- linear equation representing the field amplitude E as follows: ship between superfluid and laser dynamics was established. In view of this common theoretical description, it was expected ∂E 2 2 2 2 ¼ðD − 1ÞE − iðβ − d∇ ÞE − gðβ − d∇ Þ E − EjEj ; that the dynamics of pattern formation in lasers and the dynam- ∂τ ics of superfluidity would show identical features. Then, in the (2) 57–60 61–64,66 late 1980s and 1990s, the formation of optical trans- verse modes began to be an interesting topic for scientists in where τ is the scaled time. Equation (2) retains all the ingre- MB equation field theory. dients of spatial pattern formation in lasers. One important prop- It was found that if the Maxwell equations and the erty of the radiation in lasers is its diffraction, which is explained Schrödinger equation are coupled to constrain the N atoms by the second term on the right-hand side of Eq. (2). The third in the cavity and expand the field in the cavity mode, then term on the right-hand side of Eq. (2) describes the spatial fre- the amplitude E is coupled with atomic polarization P and quency (transverse mode) selection, a phenomenon essential for population D. The set of equations is called the MB equation the correct description of narrow-gain-line lasers. In many such system, which is also the earliest manifestation of passive lasers, the selection of transverse modes is possible by tuning systems. When extended to the laser system, the dynamic the length of the resonator. Then, the first and last terms on of the electromagnetic field in the cavity with a planar end mir- the right-hand side of Eq. (2) give the normal form of a super- ror should be considered, and the planar end mirror accommo- critical Hopf bifurcation. When the control parameter D − 1 dates two-energy-level atoms as an active medium. The form of goes through zero, a bifurcation occurs, characterized by a fixed the MB equation becomes amplitude but an arbitrary phase. For class B lasers (e.g., ruby, Nd, and CO ): γ ≫ γ ≈ κ, 2 ⊥ ∏ ∂E only the polarization intensity can be adiabatically eliminated. ¼ −ðiω þ κÞE þ κP þ idκ∇ E ∂t Then, its dynamic behavior is described by two coupled non- ∂P ¼ −γ P þ γ ED ⊥ ⊥ ; (1) ∂t linear equations corresponding to the light field and the popu- h i > 76 ∂D 1 lation as follows: ¼ −γ ðD − D Þþ ðE P þ P EÞ ∏ 0 ∂t 2 ∂E 2 2 2 ¼ðD − 1ÞE − iðβ − d∇ ÞE − gðβ − d∇ Þ E where E is the field amplitude; P is the atomic polarization ∂τ : (3) ∂D 2 intensity; D is the population intensity; κ, γ , and γ are the ¼ −γ ½ðD − D ÞþjEj D ⊥ ∏ ∏ 0 ∂τ corresponding relaxation rates; ω is the cavity resonance frequency; and d is the diffraction coefficient. The system of The first equation of Eq. (3) is the CSH dissipation equation Eq. (1) determines the behavior of the restricted electromagnetic suitable for class B lasers, where the higher-order diffusion term field E in the transverse plane, which explains the formation explains the choice of transverse modes. Comparing Eqs. (2) mechanism of the transverse mode at the atomic level. Here, and (3), it can be found that in Eq. (3), the population D is ðx; yÞ is a plane perpendicular to the z axis of the cavity, and the recovery variable and the fast light field E is the excitable it is assumed that E and P have the best plane wave dependence variable. The overall particle inversion speed is slow, and the in the z direction and the slow residual dependence on the lateral CSH equation contains additional nonlocal terms responsible variables x and y. They directly follow the Maxwell equations of for spatial mode selection, which will lead to the instability of the field E together with the Bloch equations of complex atomic pattern formation. polarization P and population N. Numerically solving the MB For class A and class B lasers, the output is stable in the ab- equations, the laser transverse patterns observed in solid-state sence of external disturbances. To realize an unstable operation, lasers are shown in Figs. 2(a) and 2(b). at least one degree of freedom needs to be added. The usual 155–157 By observing the form of the MB Eq. (1), it can be found that methods are as follows: (1) modulate a certain parameter, they are similar to the Lorentz model describing hydrodynamic such as the external field, pump rate, or cavity loss, to make instability. The similarity between the Lorentz model and the system a non-self-consistent equation system [Fig. 2(c)]; the MB equations implies that chaotic instability can happen in (2) inject the external field to increase the degrees of freedom single-mode and homogeneous line lasers. However, the consid- of the system; (3) apply a two-way ring laser. In this case, the eration of time scale excludes the complete dynamics of Eq. (1) two backpropagating modes are coupled to each other, so that in lasers. In the Lorentz model, the damping rates differ by the number of degrees of freedom of the system is greater than 1 order of magnitude from each other. On the contrary, in most two. Moreover, under the same conditions, observing the pattern lasers, the three damping rates of the MB equations are different formation of class B lasers will reveal two phenomena: periodic from each other. Then, according to the relationship among the dynamics and low-dimensional deterministic chaos [Fig. 2(e)]. three damping rates κ, γ , and γ , the MB equations can be Accordingly, fixed patterns and weak turbulence are the char- ⊥ ∏ transformed into different forms under specific laser conditions, acteristics of class A laser output patterns [Fig. 2(d)]. Advanced Photonics Nexus 024001-4 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 2 Pattern formation and time-domain properties of classes A and B lasers in specific cases. (a) Laser transverse patterns are obtained by numerically solving the MB equations. Adapted from Ref. 84. (b) Laser OL patterns with different Fresnel numbers. Adapted from Ref. 85. (c) 1D non- stationary periodic (c1) and chaotic (c2) pattern in class A lasers when the pump gain is too high. Adapted from Ref. 74. (d) 2D stationary pattern in class A lasers. (e) 2D transient nonstationary pattern (e1) and time-averaged stationary pattern (e2) in class B lasers. (d)–(e) Adapted from Ref. 76. The corresponding principle between (e1) and (e2) was discussed in Ref. 77. (f) Different time-domain properties of laser pattern formation: self-modulated periodic oscillation (f1), self- modulated quasi-periodic oscillation (f2), chaotic pulsing (f3), and single-mode stable pattern (f4). Adapted from Ref. 153. In addition, Huyet et al. used multi-scale expansion to de- organization of spatial patterns of structured laser beams. The rive the CSH equation of the laser. They obtained two fields’ formation of one-dimensional (1D) and two-dimensional (2D) equations: one is mainly caused by the phase fluctuations of spatial patterns can be explained by solving the MB equations the KS equation called the turbulent state, while the other and its modified CGL, CSH, and KS equations under specific CSH equation produces periodic modulation in spatial and conditions. In addition, since the MB equations are a system of temporal intensities. The reason that the laser intensity is locally multivariate nonlinear equations in space and time, the time- chaotic is explained by this system of equations, while the time- domain properties of certain cases in laser pattern formation, averaged intensity pattern maintains the overall symmetry of the such as stability, oscillation, and chaos, can also be described. system. The time-domain dynamics of laser pattern formation 153,159 were further studied by Chen and Lan. The ring beam 3 Eigenmode Superposition-Based Pattern distributed pumping technology is used to obtain the high-order Formation LG pattern. By slightly adjusting the spherical output cou- 0;l pling mirror, i.e., controlling the frequency difference ΔΩ of Since structured laser beams often appear as time-averaged pat- 76,77,86 the two LG patterns, the relationship between ΔΩ and the terns in practical applications, their spatial characteristics 0;l relaxation oscillation frequency ω of the solid-state laser is have received much attention in recent years. Here, we should 90–96 constantly changing, accounting for the different time-domain refer to the Helmholtz equation, which is the basic wave properties [Fig. 2(f)]. equation that the electric vector of optical frequency electromag- In summary, the derivation and deformation of the MB equa- netic field should satisfy under the scalar field approximation. tions have laid the physical foundation for the spontaneous Generally, both the Helmholtz equation and MB equations can Advanced Photonics Nexus 024001-5 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… describe the formation dynamics of laser spatial patterns, while when solved in cylindrical coordinates, is the Bessel modes, as 92,167,168 the difference is that the former does not contain the temporal shown in Fig. 3(e). If solving the Helmholtz equation in term, and the latter is a set of spatiotemporal equations. Hence, elliptical cylindrical coordinates, Mathieu–Gauss beams can be 169–171 for the analysis of spatial characteristics of laser transverse pat- obtained, as shown in Fig. 3(d). Mathieu–Gauss beams are terns, it is more common to solve the Helmholtz equation, also one class of “nondiffracting” optical fields, which are a variant of the superposition of Bessel beams. Therefore, they 2 2 have a similar capability of self-reconstruction after an opaque ∇ E þ k E ¼ 0; (4) finite obstruction. Another important solution to the paraxial 172,173 where ∇ is the Laplacian operator, E refers to the light field, wave equation is the Airy beam, as shown in Fig. 3(f). and k ¼ 2π∕λ is the light-wave number. Similar to the Bessel modes, Airy beams exhibit unique proper- ties of self-acceleration, nondiffraction, and self-reconstruction. Next, when solving the Helmholtz equation in parabolic 3.1 Eigenmodes of Helmholtz Equation coordinates, parabolic beams can be obtained, as shown in Fig. 3(g). Their transverse structures are described by parabolic The Gaussian beam is a special solution of the Helmholtz cylinder functions, and contrary to Bessel or Mathieu beams, equation under gradually varying amplitude approximation, 96 their eigenvalue spectra are continuous. Any nondiffracting beam which can describe the properties of laser beams well. The can be constructed as a superposition of parabolic beams, since Helmholtz equation can then be solved for several structured they form a complete orthogonal set of solutions of the laser beams based on the Gaussian beam. First and foremost, Helmholtz equation. Based on parabolic beams, a new family by solving the Helmholtz equation in the paraxial form, i.e., of vector beams that exhibit novel properties is explored, i.e., the paraxial wave equation, the well-known HG and LG modes parabolic-accelerating vector beams, as shown in Fig. 3(h). can be solved in Cartesian coordinates and cylindrical coordi- 162,163 This set of beams obtains the ability to freely accelerate along nates, respectively, as shown in Figs. 3(a)–3(b). The IG parabolic trajectories. In addition, their transverse polarization modes are also an important family of orthogonal solutions distributions only contain polarization states oriented at exactly to the paraxial wave equation, which represents a continuous 164–166 the same angle, but with different ellipticity. To sum up, the transition from LG to HG modes, as shown in Fig. 3(c). above-solved patterns we introduced are the eigenmodes of Another set of solutions to the Helmholtz equation in free space, Helmholtz equations, with some examples shown in Fig. 3. 3.2 Eigenmodes Superposition Theory For eigenmode superposition theory, it was found that the LG mode can be generated by coherent superposition of the HG modes as early as 1992, as shown in Fig. 4(a). It can be con- veniently understood by illustrating the above basic modes on a Bloch sphere, analogous to the Poincaré sphere (PS) for polarization, but for spatial modes. For example, if such a Bloch sphere is constructed with the LG modes, such as LG and p 0,1 LG on the poles, then the equator will represent superposi- 0;−1 tions of such beams: the HG modes. Due to the fact that these set of modes form an infinite basis, it allows one set of modes to be represented in terms of the other. Using relations between 90,180 Hermite and Laguerre polynomials, mþn ðn−k;m−kÞ ð2iÞ P ð0ÞH ðxÞH ðyÞ nþm−k k k¼0 m n−m n−m 2 2 ð−1Þ m!ðx þ iyÞ L ðx þ y Þ for n ≥ m mþn ¼ 2 × ; n m−n m−n 2 2 ð−1Þ n!ðx − iyÞ L ðx þ y Þ for m>n (5) k k ð−1Þ d ðn−k;m−kÞ n m P ð0Þ¼ ½ð1 − tÞ ð1 þ tÞ j ; (6) k t¼0 k k 2 k! dt n−m where H ð·Þ and L ð·Þ are Hermite and Laguerre polyno- k m mials, respectively, then LG modes could be represented in terms of superpositions of HG modes as Fig. 3 Basic types of transverse patterns. (a) HG beam. (b) LG mþn beam. (c) IG beam. (d) Mathieu beam. (e) Bessel beam. (f) Airy LG ðx; y; zÞ¼ ðiÞ bðn; m; KÞ · HG ðx; y; zÞ; p;l mþn−K;K beam. (a)–(f) are adapted from Ref. 42. (g) Parabolic beam. K¼0 Adapted from Ref. 160. (h) Parabolic-accelerating vector beam. (7) Adapted from Ref. 161. Advanced Photonics Nexus 024001-6 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 4 Multiple types of transverse patterns after superposition. (a) Examples of the LG modes superposed of the HG modes. Adapted from Ref. 50. (b) HLG modes in PS. Adapted from Ref. 174. (c) The intensity distribution of HLG modes with different values of α. Adapted from Ref. 175. (d) The intensity distribution of even odd IG mode with p ¼ 5 and m ¼ 3, and the HIG mode generated by the corresponding superposition when ε ¼ 0 → ∞. Adapted from Ref. 176. (e) The vortex SU(2) geometric modes. Adapted from Ref. 177. (f) The intensity and phase distribution of SHEN modes for ðn; mÞ¼ð0,6Þ. Adapted from Ref. 178. Advanced Photonics Nexus 024001-7 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 1∕2 Another similar superposition of the IG modes can form ðN − KÞ!K! 1 d n m bðn:m; KÞ¼ ½ð1 − tÞ ð1 þ tÞ j ; t¼0 N K patterns with multi-singularity, named the helical IG (HIG) 2 n!m! K! dt 186–188 modes. HIG modes are obtained by coherent superposition (8) of even and odd IG modes, whose expression is where l ¼ m − n, p ¼ minðm; nÞ, and N ¼ m þ n. This leads e o HIG ¼ IG ðξ; η; εÞ iIG ðξ; η; εÞ; (11) p;m p;m p;m to an alternative description of light fields through the perspec- tive of mode superposition, particularly useful in the description where is the direction of the vortex; p and m are the orders of of beams with a transverse profile that is invariant during propa- the IG mode; o and e represent the odd mode and the even gation. For the mutual superposition and conversion of HG mode, respectively; ε is the ellipticity parameter, indicating and LG eigenmodes, Abramochkin and Volostnikov in 2004 in- the change degree of ellipticity. ξ and η are elliptic coordinates; troduced a parameter α to unify them. Such beams with more e;o and IG ð·Þ represent the expression of IG modes. When universality are called generalized Gaussian beams or Hermite– p;m e;o 41,175,182,183 ε → 0, IG modes could be reduced to LG modes with Laguerre–Gaussian (HLG) beams. The expression of p;m p:l e;o HLG beam is as follows: l ¼ m and p ¼ 2n þ l. When ε → ∞, IG modes could be p;m reduced to HG modes with n ¼ m − 1 and n ¼ n :n x y x y p − m þ 1. Therefore, for ε ¼ 0 → ∞, the HIG modes com- 1 jrj r HLG ðr; zjαÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp −π HL pffiffiffi jα n;m n;m posed of even and odd modes show specific distribution N−1 w πw 2 n!m! changes, shown in Fig. 4(d). Comparing HLG and HIG modes, it could be found that they × exp ikz þ ik − iðm þ n þ 1ÞΨðzÞ ; all possess phase singularities with corresponding spatial distri- 2R butions for differently composed HG and LG modes. The above (9) HG, LG, HLG, and HIG modes belong to the general family of structured Gaussian modes, also known as the singularities where HL ð•Þ are Hermite–Laguerre (HL) polynomials, r ¼ n;m hybrid evolution nature (SHEN) modes. The model expres- T T 2 2 2 ðx; yÞ ¼ðr cos ϕ;r sin ϕÞ , RðzÞ¼ðz þ z Þ∕z, kw ðzÞ¼ sion of SHEN modes is 2 2 2ðz þ z Þ∕z , ΨðzÞ¼ arctanðz∕z Þ, and z is the Rayleigh R R R range. It was found that when α ¼ 0 or π∕2, the mode HLG will be reduced to HG or HG mode. When α ¼ n;m n;m m;n iβK SHEN ðx; y; zjβ; γÞ¼ e bðn; m; KÞ; n;m π∕4 or 3π∕4, the mode HLG will be reduced to LG mode n;m p;l K¼0 ½p − minðm; nÞ;l ¼ m − nÞ. For other α, the mode HLG is n;m K e 2 K ð−iÞ IG ðx; y; zjε ¼ 2∕tan γ; for ð−1Þ ¼ 1 N;N−K displayed as the beam distribution of multi-phase singularities. • : K o 2 K Take HLG mode as an example, when α ¼ 0, π∕4, and π∕8, ð−iÞ IG ðx; y; zjε ¼ 2∕tan γ; for ð−1Þ ≠ 1 3,1 N;N−Kþ1 the mode HLG could be reduced to HG , LG , and mode 3,1 31 12 175 (12) with multi-phase singularities, as shown in Fig. 4(c). Since HLG modes are able to unify HG modes and LG modes, it When β ¼π∕2, SHEN modes could be reduced to HIG was mapped in the PS to show the relationship among HLG, modes. When γ ¼ 0, SHEN modes could be reduced to HLG HG, and LG modes. As shown in Fig. 4(b), the poles represent modes. When ðβ; γÞ¼ð0,0Þ or ðπ; 0Þ, SHEN modes could be the high-order LG modes with opposite topological charges, reduced to HG modes. When ðβ; γÞ¼ ðπ∕2,0Þ, SHEN modes and the equator represents the high-order HG modes, while the could be reduced to LG modes. Therefore, SHEN modes can mode between the poles and the equator represents the transi- uniformly describe HG, LG, HIG, and HLG modes. In addition, tional high-order HLG modes. the PS could be applied to define the SHEN sphere to describe On the basis of HLG modes, analyzing its coherent superpo- these modes, as shown in Fig. 4(f). sition can obtain a high-dimensional complex light field in a It can be seen that the above patterns are derived from the coherent state, whose typical type is SU(2) mode. The SU(2) direct coherent superposition of the basic modes. The beam mode appears when the laser mode undergoes frequency degen- generated after superposition can be regarded as a new kind eracy with a photon performing as an SU(2) quantum coherent 174,177,184,185 of eigenmodes with single-frequency operation. If performing state coupled with a classical periodic trajectory, coherent superposition between these eigenmodes, diverse and which contains both spatially coherent wave packets and geo- complex structured laser beam patterns could be generated. metric ray trajectories, as shown in Fig. 4(e). Taking the HLG Moreover, in a practical resonator, if multiple modes interact mode as the basic mode of SU(2) coherent state, the superposed coherently, the coupling of frequency and phase will be formed SU(2) mode is expressed as spontaneously to achieve TML. The principle of the TML 97–101 effect includes frequency locking and phase locking. The N 1 1 2 N;P;Q iKϕ total electric field of a beam in the coherent superposition state jψ i¼ e jψ i; (10) nþQK;m;l−PK n;m:l N∕2 99 can be given as K¼0 where ϕ is the phase term, and P∕Q (P and Q are coprime in- E ¼ a XG ð·Þ tot m;n m;n tegers) is the ratio of the distance between the transverse mode m;n and longitudinal mode leading to frequency degeneracy, i.e., the 2 2 x þ y transverse mode and longitudinal mode of various eigenmodes exp iϕ þ ikz þ ik − iqψðzÞ ; (13) m;n RðzÞ should meet the coherent superposition condition. Advanced Photonics Nexus 024001-8 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… where XG represents the polynomial of eigenmodes, such as interval between two neighboring vortices should also be HG, LG, and IG. The subscripts m and n are the order indices small (usually tens of micrometers) for the formation of stable of the corresponding basic modes, a is the weight of each patterns. Fourth, a wide cavity without mechanical boundaries m;n basic mode, expð…Þ is the phase item, ϕ is the initial phase, is also needed. The patterns produced by the eigenmode m;n and qψðzÞ is the Gouy phase. coherent superposition in the TML state include optical lattice 86,88,89 The spatial pattern is analyzed by superimposing the electric (OL) and OVL, which could be obtained in vertical- field of multiple modes with the locking phase, including the cavity surface-emitting lasers (VCSELs) and solid-state la- 189 88,89,99,192 Gouy phase. Moreover, with the assistance of the inherent sers, as shown in Figs. 5(a)–5(e). In addition to coherent superposition, incoherent superposition can also produce pat- nonlinearity of the laser cavity, the frequencies of the composed 193,195 87–89 terns similar to OL, as shown in Figs. 4(f) and 4(g). The modes are possible to be pulled to the same value, and then incoherent superposition is to analyze the output OL pattern the total electric field can be expressed as through the superposition of intensities of the composed modes 2 2 alone. It is pointed out that in multi-transverse-mode lasers, the ω ω x þ y E ðx; y; zÞ¼ exp i z þ i − iqψðzÞ tot coupling between transverse modes occurs through their inten- c c RðzÞ sities rather than their field amplitudes, and these modes are arranged according to the principle of transverse hole burning · a XG ð·Þ expðiϕ Þ; (14) m;n m;n m;n to maximize energy coexistence and minimize intensity distri- m;n bution overlap. It is found that the beam patterns generated by incoherent superposition have higher symmetry, and there may where ω is the averaged optical frequency, whose derivation is be no phase singularity in some dark areas of the pattern. as follows: We have summarized the principle of directly generating spatially structured beams from the laser cavity, namely, the a Δω > m;n n m;n < P ω ¼ ω þ 0 eigenmode superposition theory. The spatial characteristics of m;n m;n : (15) 2 2 the spontaneous organized patterns are explained by the inter- cðk þk Þ x y Δω ¼ ω − ω ¼ n n 0 action and superposition of the oscillation modes. However, 2k with the help of some modulation devices, such as the spiral 196–198 199 200 Here, ω is the optical frequency of the fundamental trans- phase plate, diaphragm, acousto-optic modulator, liquid 201–205 145 206–208 verse mode, Δω is the frequency spacing between the nth crystal Q-plate, J-plate, liquid crystal SLM, and 209–211 mode and the fundamental mode, c is the velocity of light, digital micromirror device, various spatial structure laser k ¼ πn∕l , k ¼ πn∕l , k ¼ 2π∕λ, and l and l are the sizes beams can also be generated indirectly. There are also many x x y x z x y 40,42,43,45–47,50,212 of the cavity in the x and y directions, respectively. Along with reviews on this part of active regulation to generate the locking of frequencies, the parameters of RðzÞ and index of spatially structured laser beams, and more detailed information can be found in those reviews. ψðzÞ should also be an averaged one to help with the locking of the total phases. The cooperatively frequency-locked multimode regime, in which at least two transverse modes contribute sig- 4 Spatiotemporal Beam Patterns nificantly to the output field, and lock to a common frequency As for the study of structured laser pattern formation, the with which they oscillate in a synchronized way. The locking traditional electromagnetic field equations including MB and concerns also the relative phases of the modes, so that the output Helmholtz equations were adopted in recognition of the impor- intensity has a stationary transverse configuration. The common tance of gain and loss. In addition, the eigenmode superposition oscillation frequency, cooperatively selected by the modes, cor- theory makes it possible to form diverse and complex beam responds to the average of the modal frequencies, weighted over patterns. When the laser oscillates simultaneously in multiple the intensity distribution of the modes in the stationary state. modes and the phase difference between them is stable, the The patterns formed by TML effect could possess phase sin- mode locking occurs. What we talked about the spatial charac- gularities in dark points. Around each of these phase singu- teristics of laser transverse modes in Sec. 3 is to study the TML larities, the modulus of the electric field raises from zero in the under the condition of single longitudinal mode. However, form of an inverted cone with a steep gradient. If performing a if multiple longitudinal modes are involved, total mode locking 103–106 closed counterclockwise loop that surrounds one of these points, or spatiotemporal locking will occur, thus generating the phase of the envelope of the electric field changes by a value spatiotemporal laser beams. equal to 2πm, where m is a positive integer. These properties are fulfilled also in the “optical vortices” discovered by Coullet 4.1 Spatiotemporal Mode Locking 63 62 and collaborators in their 2D analysis of the model. A major difference is that from the fact that the vortices in Ref. 63 can be The spatiotemporal mode locking is often realized by fiber generated in any position of the transverse plane, whereas lasers, referring to the coherent superposition of longitudinal the singularities that appear in the stationary configurations of and transverse modes of the laser, which allows locking a laser system are located in precisely defined positions, i.e., the multiple transverse and longitudinal modes to create ultrashort dark points. pulses with various spatiotemporal distributions, as shown in The stable beam pattern formed by coherent superposition Figs. 6(a1) and 6(a2). The locking of transverse and longitudinal through TML usually needs to meet the following conditions: modes of a laser is realized by spatial filtering and spatiotem- 116,117 First, there should be a large value of the Fresnel number to poral normal dispersion mode locking, respectively. The sustain such multi-transverse modes. Second, the transverse spatiotemporal mode locking can be realized through the high mode spacing, Δv , should be small (several gigahertz or lower) nonlinearity, gain, and spatiotemporal dispersion of the optical 190 103 to assist nonlinear coupling in the spectral band. Third, the fiber medium, as well as spectral and spatial filtering. These Advanced Photonics Nexus 024001-9 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 5 Multiple patterns produced by coherent and incoherent superposition. (a)–(e) OVL patterns from coherent superposition of HG, LG, and IG modes, while (a) is from VCSELs. Adapted from Ref. 87. (b) Beam patterns from a solid-state LNP laser. Adapted from Ref. 88. (c) Beam patterns from a solid-state Yb:CALGO laser. Adapted from Ref. 89. (d) Beam patterns from a microchip Nd:YAG laser. Adapted from Ref. 99. (e) Beam patterns from a solid-state Pr:YLF laser. Adapted from Ref. 192. (f)–(g) Beam patterns from incoherent superposition of the (f) LG and (g) HG modes in solid-state lasers. Adapted from Refs. 193 and 194. Advanced Photonics Nexus 024001-10 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 6 Spatiotemporal mode-locking beam patterns. (a) Spatiotemporal mode locking through both longitudinal and transverse modes. Adapted from Ref. 103. (a1) Transverse distributions. (a2) Pattern, spectra, and intensity of composed modes. (a3) Schematic diagram of the cavity sup- porting spatiotemporal mode locking. (b) Experimental regimes of spatiotemporal mode locking and results from a reduced laser model. Adapted from Ref. 107. (c) Phase locking of the longitudinal and transverse (TEM and TEM ) modes to create scanning beam. Adapted from Ref. 114. 00 01 Advanced Photonics Nexus 024001-11 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… inseparable coupling effects can be described by the cavity pump laser end mirror to induce time-varying pulse coupling operator C as follows: in a photonic crystal fiber for subsequent generation of fre- quency-shifted Raman solitons. The introduced spatial oscilla- ˆ ˆ ˆ c ˆ tions of the output beam lead to modulation of the coupling C ¼ Fðx; yÞFðωÞSAðx; y; tÞPðx; y; tÞ; (16) efficiency of the fiber and effectively induces wavelength ˆ ˆ sweeping. where Fðx; yÞ and FðωÞ are the spectral and spatial filter func- tions, respectively, SAðx; y; tÞ is the spatiotemporal saturable absorber transfer function, and Pðx; y; tÞ accounts for the effect 4.2 Spatial Modulation of Mode-Locked Laser Pulses of the pulse propagation through the three-dimensional (3D) The spatiotemporal mode-locked pulses directly produced by nonlinear gain medium. P includes the inseparable effects of fiber lasers are often irregularly distributed. Another method 3D gain, such as spatiotemporal dispersion and nonlinear mode to generate spatiotemporal beams with regular and complex coupling. With the composition of iterated nonlinear projection distribution is to use the pulse-shaping device based on SLM. operations, the field of spatiotemporal locking pulse can be The designed pulse shaper is usually applied to shape the input expressed as femtosecond laser to obtain the specific spatiotemporal pattern. As shown in Fig. 7(a), the spatiotemporal optical vortices have E ðx; y; tÞ¼ CE ðx; y; tÞ; (17) iþ1 i been proved to be generated using spiral phase in the pulse shaper. Suppose an optical field in the spatial frequency– where the subscript of E is the round-trip number. In each round frequency domain (k − ω) is given by g ðrÞ. After a spiral ˆ x R trip, the nonlinear dissipation of C is selected from the field −ilθ phase of e is applied, a 2D Fourier transform gives the field certain attributes, and the saturable laser gain provides a condi- in the spatial-temporal ðx; tÞ domain as follows: tional (frequency and energy-limited, and spatially localized) rescaling of the selected field. −ilθ l −ilθ The schematic diagram of the cavity supporting spatiotem- Gðρ; ϕÞ¼ FTfg ðrÞe g¼ 2πð−iÞ e H fg ðrÞg; (18) R l R poral mode locking is shown in Fig. 6(a3). The fiber ring laser pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi was composed by offset splicing a graded-index fiber to a few- 2 2 where ðr; θÞ are the polar coordinates with r ¼ k þ ω and mode (three modes were supported) Yb-doped fiber amplifier, −1 θ ¼ tan ðω∕k Þ, and ðρ; ϕÞ are the Fourier conjugate polar pffiffiffiffiffiffiffiffiffiffiffiffiffiffi which leads to spatial filtering action. Spatial filtering can 2 2 −1 coordinates with ρ ¼ x þ t and θ ¼ tan ðx∕tÞ. Here, lock multiple transverse modes by controlling the overlap of H fg ðrÞg ¼ rg ðrÞJ ð2πρrÞdr and J is the Bessel func- l R R l l fields coupled to the optical fiber. Then, the self-starting mode tion of the first kind. locking in the normal chromatic dispersion regime is achieved Therefore, starting from chirped mode-locked pulses, a using a combination of spectral filtering and intracavity nonlin- diffraction grating and cylindrical lens disperse frequencies ear polarization rotation, which is realized in the longitudinal spatially and act as a time-frequency Fourier transform. Then, mode locking. Through the space–time locking of the transverse a spiral phase on the SLM and an inverse Fourier transform and longitudinal modes, ultrashort pulses with special space– by recollecting dispersed frequencies with a grating-cylindrical time distribution can be obtained. In the process of spatiotem- lens pair form the chirped spatiotemporal vortex. The full elec- poral mode locking, the mode dispersion will affect the locking tric field is presented as effect, and the small mode dispersion of graded-index multi- mode fiber is considered to be a key factor to make spatiotem- Eðρ; ϕÞ¼ Gðρ; ϕÞ expðik z − iωtÞ: (19) poral mode locking possible. Experimental regimes of spatiotemporal mode locking and results from a reduced laser After generating the spatiotemporal vortex pulse, it travels model are shown in Fig. 6(b). The reduced models predicted through an afocal cylindrical beam expander and stretches in how the effects of disorder, and the increased dimension of the direction of the vortex line. It is reported that the stretched the optimization, affect the regimes of spatiotemporal mode spatiotemporal vortex pulse could transform into a toroidal vor- locking. On this basis, it was found that spatiotemporal mode tex pulse through a conformal mapping system formed by two locking can also be realized in multimode fiber lasers with large mode dispersion, in which the intracavity saturable SLMs, as shown in Fig. 7(b). In addition, the spatiotemporal absorber plays an important role in offsetting the large mode optical vortex was also demonstrated to be generated from a dispersion. Spatiotemporal mode locking at a fiber laser using light source with partial temporal coherence and fluctuating a step-index few-mode thulium fiber amplifier and a semicon- temporal structures. Similarly, through a phase mask in SLM, ductor saturable absorber was also reported. The former real- interesting wave packets, such as diffraction-free pulsed beams izes spatial filtering to lock the transverse mode, and the latter with arbitrary 1D transverse profiles without suffering power plays a role in longitudinal mode locking. loss were generated. The basic concept [illustrated in Fig. 7(c)] Therefore, spatiotemporal mode locking is affected by gain, combines spatial-beam modulation and ultrafast pulse shaping spatial filtering, optical nonlinear interaction between saturable and is related to the so-called 4f-imager used to introduce absorbers, and optical fiber medium, as well as the coupling be- spatiotemporal coupling into ultrafast pulsed beams. The modu- tween temporal and spatial degrees of freedom. Apart from lated beam is reflected back, and the pulse is reconstituted by the multimode fiber lasers, in all few-mode fiber, it is realizable grating to produce the spatiotemporal light sheet. Additionally, to obtain spatiotemporal mode locking to create bound-state it applied reflective annular mask in a pulse shaper to generate a 109 122 solitons. In addition, the scanning output beam can be gener- spatiotemporal Bessel wave packet, as shown in Fig. 7(d). ated by spatiotemporal locking of the laser mode, as shown in Since the mask has an annular shape, it is then possible to obtain Fig. 6(c). The phase locking of the longitudinal and transverse a spatiotemporal Bessel beam at its waist by doing the inverse (TEM and TEM ) modes is simply obtained by tilting the Fourier transform in the space and time domains of the signal 00 01 Advanced Photonics Nexus 024001-12 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 7 Spatiotemporal beam patterns generated by a pulse shaper. (a) Generation (a1) and measurement (a2) of the spatiotemporal vortex. Adapted from Ref. 118. (b) Generation of the spatiotemporal toroidal vortex. Adapted from Ref. 126. (c) Generation of spatiotemporal Airy beams. Adapted from Ref. 121. (d) Generation of spatiotemporal Bessel beams. Adapted from Ref. 122. (e) Schematic of a device capable of mapping an input vector spatiotemporal field onto an arbitrary vector spatiotemporal output field. Adapted from Ref. 125. Advanced Photonics Nexus 024001-13 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… reflected by the mask. This analysis has revealed that these the z direction with frequencies ω , ω , and ω , respectively. 1 2 3 beams are produced by a superposition of plane waves of differ- Considering only the second-order nonlinear effect, the second- ent optical frequencies and directions of propagation traveling at order nonlinear polarization can be written as the same group velocity along the z axis. With all plane waves added in the same phase, a nondiffraction spatiotemporal beam ð2Þ ð2Þ P ðω Þ¼ 2ε χ ð−ω ; ω ; −ω Þ · E E < 1 0 1 3 2 3 can be generated through spatiotemporal coupling. ð2Þ ð2Þ P ðω Þ¼ 2ε χ ð−ω ; ω ; −ω Þ · E E ; (21) 2 0 2 3 1 3 In addition, a device for generating an arbitrary vector spa- ð2Þ ð2Þ P ðω Þ¼ 2ε χ ð−ω ; ω ; ω Þ · E E tiotemporal light field with arbitrary amplitude, phase, and 3 0 3 1 2 1 2 polarization at each point in space and time was designed. As shown in Fig. 7(e), the laser output with two orthogonal and we can get the coupled wave equations for three-wave polarizations propagates through the pulse shaper to redistribute interaction as between the space domain and the time domain. The shaped light propagates through the multi-plane light conversion dE ðzÞ iω 1 1 ð2Þ iΔkz ¼ χ ð−ω ; ω ; −ω Þ · E ðzÞE ðzÞe 1 3 2 3 > dz cn (MPLC) device and is converted to different HG modes at dE ðzÞ iω 2 2 ð2Þ iΔkz the output port of the MPLC. Then, arbitrary spatiotemporal ¼ χ ð−ω ; ω ; −ω Þ · E ðzÞE ðzÞe : (22) 2 3 1 3 dz cn beams can be generated through the design and combination dE ðzÞ iω ð2Þ iΔkz : 3 3 ¼ χ ð−ω ; ω ; ω Þ · E ðzÞE ðzÞe 3 1 2 1 2 of HG beams at different times. These methods of transforming dz cn spatiotemporal beams through optical devices, such as pulse shapers, gratings, and lenses can effectively generate spatiotem- Then, the field intensity of a specifically structured light poral beams with specific structures. They can be used in the beam after nonlinear transformation can be obtained by substi- fields of imaging, optical communication, nonlinear optics, tuting the structured light field expression XG (e.g., HG, LG, particle manipulation, and so on. and IG) into the coupled wave Eq. (22) and solving them. 5.1 External Cavity Nonlinear Processes 5 Structured Beam Patterns Generated by Nonlinear Processes External cavity pattern modulations and nonlinear interactions are the main form of nonlinear transformation in structured laser For the booming research on the spatial and spatiotemporal beams. The first research on the transformation of structured properties of structured laser beams reviewed in the above sec- laser beams in nonlinear optics began with the SHG of LG tions, investigations are based on beams at a single wavelength. modes in 1996. It was found that the SHG fields carry Currently, the nonlinear transformation technology for funda- twice the azimuthal indices of the pump, which provided mental mode Gaussian beams is very mature. The combination straightforward insight into OAM conservation during nonlinear between structured laser beams and nonlinear transformation on interactions at the photon level. The relationship among the the transverse pattern variation has been of great interest in 127–130 131–140 141,142 OAMs of the input l ;l ; … and output beams l was recognized recent years. The SFG, SHG, FWM, and other 1 2 143–145 to follow the law of l ¼ l þ l þ … þ l . That is, OAM is con- frequency upconversion methods have been studied in the 1 2 n served in the nonlinear process. As such, for SHG modes of two nonlinear process of structured laser beams. The variation of photons with angular frequency ω and a single OAM, if the OAM in the nonlinear process are one of the focuses of these OAMs are equal, it has l ¼ 2l or l ¼ l þ l ″, as shown studies. In this section, the nonlinear process of structured laser 2ω ω 2ω ω ω in Fig. 8(a). This conservation law is also applicable to frac- beams is introduced from two aspects: external-cavity modula- tional and odd-order OAM beams. The generation of LG tions and intracavity transformation. We will first present the beams with higher radial orders and IG beams through a non- physical mechanisms. linear wave mixing process was further investigated to obtain Generally, the nonlinear transformation of structured patterns 130,139,140 more complex beam patterns. However, in the above is based on the nonlinear wave equation, studies, the characteristics of the frequency-doubled beam patterns during propagation have hardly been mentioned. Later, ∂E iω NL iΔkz in the research of SHG modes through input LG beams with ¼ P e ; (20) ∂z 2ε cn opposite OAMs, the near- and far-field patterns show different light field distributions, as shown in Fig. 8(b). In addition, for NL where E is the electric field, ω is the optical frequency, P is the change of pattern transmission, a more detailed theoretical the nonlinear polarization, and Δk is the wave vector difference model is proposed to describe the beam pattern transmission between the polarized wave and the incident light. The nonlin- and radial mode transition in the process of SHG, and the ear process in the multi-wave mixing process can be obtained results agree well with the simulations, as shown in Fig. 8(c). by solving the coupled wave equations. Generally speaking, for Apart from the single OAM state, phase structure transfer in the n’th-order nonlinear effect, n þ 1 nonlinear coupled wave FWM and sum frequency129 processes were also investigated equations corresponding to different frequencies can be listed. for the input beams possessing coherent superposition of LG By simultaneously solving the n þ 1 coupled wave equations, beams. the electric field strengths of these different frequencies of light As shown in Fig. 8(d), there are two sets of patterns and can be obtained, thereby leading to the law of mutual conversion propagation behaviors: (1) one pump beam carries OAM super- of energy between these light fields. In the nonlinear transfor- position mode with another pump beam carrying a single OAM mation of structured patterns, the second-order nonlinear mode and (2) both pump beams carry OAM superposition effect is the main part, including SFG and SHG. Consider three modes. The relationship between the SFG beam and the pump monochromatic plane waves E , E , and E propagating in beam is as follows: 1 2 3 Advanced Photonics Nexus 024001-14 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 8 External cavity nonlinear process of structured laser beams. (a) Experimental setup and results showing the OAMs of the input and output beams are equal. Adapted from Ref. 135. (b) Experimental setup and results showing different SHG pattern distributions in near and far fields. Adapted from Ref. 137. (c) SHG patterns with beam pattern transmission and radial mode transition. Adapted from Ref. 145. (d) Experimental setup and results of SFG modes with input beams possessing coherent superposition of LG beams. Adapted from Ref. 129. Advanced Photonics Nexus 024001-15 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… l iθ −l l 2l iθ 0 As shown in Figs. 9(d2)–9(d3), a set of TML beams composed > ðLG þ e LG Þ LG → LG þ e LG 0 0 0 0 l of the HG and LG modes and their SHG beams were obtained l −l l l þl < 1 iθ 2 2 1 2 iθ 0 ðLG þ e LG Þ LG → LG þ e LG 0 0 0 0 l experimentally and agreed well with the theoretical simulation E ∝ ; SFG l iθ −l l iθ −l 2l model as follows: 1 2 > ðLG þ e LG ÞðLG þ e LG Þ → LG > 0 0 0 0 0 iðθ þθ Þ −2l iθ iθ 0 1 2 1 2 þe LG þðe þ e ÞLG iϕ 0 l E ðr; φÞ ∝ ½XG ðr; φÞþ e XG ðr; φÞ SHG m1;n1 m2;n2 (23) iϕ × ½XG ðr; φÞþ e XG ðr; φÞ m1;n1 m2;n2 2 iϕ ¼ XG ðr; φÞþ 2e XG ðr; φÞXG ðr; φÞ where LG represents the standard LG mode, and θ is the phase m1;n1 m2;n2 m1;n1 item. It is worth noting that LG is similar to the standard LG i2ϕ 2 þ e XG ðr; φÞ: (24) 0 m2;n2 mode LG , but differs by a factor of 2 in the Laguerre polyno- mials. This difference makes the intensity in the radial direction The complex transverse patterns in the TML states were of the SFG beam decrease much more rapidly than the standard composed of different basic modes with different weight coef- LG mode. ficients and different locking phases, which makes the spatial information of the fundamental frequency mode and its SHG 5.2 IntraCavity Nonlinear Processes beam quite abundant. In summary, combining the nonlinear transformation with All these above studies on nonlinear processes were explored on the study of structured laser beams, the beam patterns are en- the basis of external cavity structured laser pattern generation. dowed with richer spatial information characteristics. For exter- Usually, in these studies, structured beams were first generated nal cavity nonlinear process of structured laser beams, the law of with the help of modulation devices (such as SLM), and then OAM conservation during nonlinear interactions of LG beams frequency conversion was carried out. For patterns generated was found. For SHG and SFG modes of the special LG beams, through intracavity nonlinear process, some studies showed new the propagation of the output beam patterns from near-field to properties. In Fig. 9(a1), it shows a frequency-doubled cavity far-field shows varying spatial characteristics. For intracavity that converts the infrared fundamental frequency of Nd:YAG nonlinear processes of structured laser beams, much more com- (λ ¼ 1064 nm) to the second-harmonic green (λ ¼ 532 nm) plex and diverse beam patterns could be obtained. In general, as through an intracavity nonlinear crystal (KTP). The concept of the laser design exploits a unique feature of OAM coupling to a new research field of structured laser beams, the nonlinear pro- linear polarization states. The resonant mode morphs from a cess of beam shaping can be widely used in 3D printing, optical linearly polarized Gaussian-like enveloped beam at one end of trapping, and free-space optical communication. In addition, as the cavity to an arbitrary angular momentum state at the other. reviewed in Secs. 3 and 4, since there are many techniques for A polarizer was required for selection of the horizontal polari- generating spatial and spatiotemporal structured beams. By ap- zation state before the J-plate, and the polarization of the light plying nonlinear transformation technology, it is expected that traversing the J-plate was controlled by simply rotating the the future of structured laser beams will have broader develop- J-plate itself. Various measured states from the laser, displayed ment prospects. on a generalized OAM sphere are shown in Fig. 9(a2). The tran- sition patterns from one to the other allow visualization of lasing 6 Conclusions and Perspectives across vastly differing OAM values as superpositions with two This paper is dedicated to reviewing the evolution of the spatial concentric rings. Then, in a digital laser for on-demand intra- patterns of structured laser beams, covering the spontaneous cavity selective excitation of second-harmonic higher-order organization of patterns described by relative equations and modes, an SLM used for structured beam generation also acted the advancements of on-demand transformations of laser pat- as an end mirror of the laser resonator, as shown in Fig. 9(b1). terns. Taking the spatial pattern as the core, we first reviewed After SHG modes passed through the nonlinear KTP crystal, the theoretical basis of laser transverse mode formation and em- it was found that the near-field spatial intensity profiles of the phasized its electromagnetic field properties and the dynamic SHG LG modes in Fig. 9(b3) are similar to the intensity profile mechanisms described by the related equations. Then, we ana- of the fundamental LG pump modes in Fig. 9(b2). But at the far field, the spatial intensity profiles of the SHG LG modes in lyzed the latest developments in the spatial characteristics of Fig. 9(b4) are different from the fundamental pump modes structured laser beam patterns through eigenmode superposition because there is an added central intensity maximum, while in theory. With the coherent and incoherent superposition of laser another intracavity SHG generation laser, the spatial distribu- eigenmodes, complex and diverse spatial patterns of structured tions of the SHG beams are almost the same as that of the laser beams can be generated. These studies on the spatial char- pump beams, as shown in Figs. 9(c2)–9(c3). For some SHG acteristics of structured laser beams are often conducted under patterns, the topological charge was found to have doubled, as the premise of a single longitudinal mode. However, if multiple shown in Fig. 9(c4). The generation of high-order modes was longitudinal modes are involved, the time dimension needs to be obtained by the off-axis displacement of the output coupling accounted for. Therefore, we later reviewed the research on mirror and the SHG process was through the intracavity BBO spatiotemporal structured laser beams, including direct genera- crystal, as shown in Fig. 9(c1). In addition, investigations on the tion by spatiotemporal mode-locking effect in fiber lasers and SHG structured laser beams in the TML states have been carried indirect regulation through the pulse shaper based on SLM. out. Through a sandwich-like microchip laser composed of Moreover, it was found that the structured laser patterns could Nd:YAG, Cr:YAG, and LTO crystals in Fig. 9(d1), complex be endowed with richer spatial information characteristics and diverse structured beams in TML states and their SHG through nonlinear conversion processes. We finally reviewed beams were generated by altering the pumping parameters. various nonlinear processes of structured laser patterns ranging Advanced Photonics Nexus 024001-16 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… Fig. 9 Intracavity nonlinear process of structured laser beams. (a) Experimental setup and results to generate intracavity frequency-doubled LG beams. Adapted from Ref. 146. (b) Experimental setup and results showing near- and far-field SHG LG beams. Adapted from Ref. 147. (c) Experimental setup and results showing SHG optical vortices. Adapted from Ref. 148. (d) Experimental setup and results showing SHG modes of structured laser beams in the TML states. Adapted from Ref. 149. Advanced Photonics Nexus 024001-17 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 19. F. Ferdous et al., “Spectral line-by-line pulse shaping of on-chip from external cavity modulations to intracavity transformation microresonator frequency combs,” Nat. Photonics 5(12), 770– comprehensively. Looking back over these 10 years, we can 776 (2011). see how much our research and understanding of laser spatial 20. A. Hugi et al., “Mid-infrared frequency comb based on a quan- features have advanced from classical to quantum, especially tum cascade laser,” Nature 492(7428), 229–233 (2012). since the discovery of OAM in the 1990s. However, the study of 21. G. Herink et al., “Resolving the build-up of femtosecond mode- structured laser beams may just be in its early stages. We still locking with single-shot spectroscopy at 90 MHz frame rate,” have many unexplored novel phenomena and theories. There is Nat. Photonics 10(5), 321–326 (2016). potential for new and improved applications based on these 22. W. Liang et al., “High spectral purity Kerr frequency comb spatial and temporal properties of laser modes, which could radio frequency photonic oscillator,” Nat. Commun. 6,7957 (2015). facilitate further research on novel laser spatial structures. 23. C. R. Petersen et al., “Mid-infrared supercontinuum covering the 1.4 − 13.3 μm molecular fingerprint region using ultra-high NA Acknowledgments chalcogenide step-index fibre,” Nat. Photonics 8(11), 830–834 (2014). The authors declare no conflict of interest. 24. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011). References 25. C. Maurer et al., “What spatial light modulators can do for optical 1. T. H. Maiman, “Stimulated optical radiation in ruby,” Nature microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011). 187, 493–494 (1960). 26. R. Heintzmann and M. Gustafsson, “Subdiffraction resolution in 2. Y. S. Kivshar et al., “Dynamics of optical vortex solitons,” Opt. continuous samples,” Nat. Photonics 3(7), 362–364 (2009). Commun. 152(1–3), 198–206 (1998). 27. J. Huisken et al. “Optical sectioning deep inside live embryos by 3. F. Encinas-Sanz, S. Melle, and O. G. Calderón, “Time-resolved selective plane illumination microscopy,” Science 305(5686), dynamics of two-dimensional transverse patterns in broad area 1007–1009 (2004). lasers,” Phys. Rev. Lett. 93(21), 213904 (2004). 28. A. Belmonte and J. P. Torres, “Optical Doppler shift with struc- 4. P. Ryczkowski et al., “Real-time full-field characterization of tured light,” Opt. Lett. 36(22), 4437–4439 (2011). transient dissipative soliton dynamics in a mode-locked laser,” 29. C. Rosales-Guzmán et al., “Experimental detection of transverse Nat. Photonics 12(4), 221–227 (2018). particle movement with structured light,” Sci. Rep. 3, 2815 5. A. Giesen and J. Speiser, “Fifteen years of work on thin-disk (2013). lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum 30. B. Aniceto et al., “A measure of flow vorticity with helical beams Electron. 13(3), 598–609 (2007). of light,” Optica 2(11), 1002–1005 (2015). 6. H. Daido, M. Nishiuchi, and A. S. Pirozhkov, “Review of laser- 31. J. Wang et al., “Terabit free-space data transmission employing driven ion sources and their applications,” Rep. Prog. Phys. orbital angular momentum multiplexing,” Nat. Photonics 6(7), 75(5), 56401 (2012). 488–496 (2012). 7. M. N. Zervas and C. A. Codemard, “High power fiber lasers: a 32. J. Wang, “Advances in communications using optical vortices,” review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 Photonics Res. 4(5), B14–B28 (2016). (2014). 33. L. Gong et al., “Optical orbital-angular-momentum-multiplexed 8. Y. Yang et al., “Whispering-gallery mode hexagonal micro-/ data transmission under high scattering,” Light Sci. Appl. 8(1), 27 nanocavity lasers [invited],” Photonics Res. 7(05), 594–131 (2019). (2019). 34. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. 9. C. Lecompte et al., “Laser temporal-coherence effects on multi- Photonics 5(6), 343–348 (2011). photon ionization processes,” Phys. Rev. A 11(3), 1009–1015 35. C. Kuo and S. Chu, “Numerical study of the properties of optical (1975). vortex array laser tweezers,” Opt. Express 21(22), 26418–26431 10. M. Skowronek, “Investigation of laser temporal pulse duration (2013). on Rayleigh scattering,” Phys. Rev. A 27(6), 3349–3350 36. W. Mike et al., “Advanced optical trapping by complex beam (1983). shaping,” Laser Photonics Rev. 7(6), 839–854 (2013). 11. V. Hohreiter, J. E. Carranza, and D. W. Hahn, “Temporal analysis 37. X. L. Wang et al., “Quantum teleportation of multiple degrees of of laser-induced plasma properties as related to laser-induced freedom of a single photon,” Nature 518(7540), 516 (2015). breakdown spectroscopy,” Spectrochim. Acta Part B At. Spectrosc. 38. B. Ndagano et al., “Creation and detection of vector vortex 59(3), 327–333 (2004). modes for classical and quantum communication,” J. Lightwave 12. O. Lundh et al., “Few femtosecond, few kiloampere electron Technol. 36(2), 292–301 (2018). bunch produced by a laser–plasma accelerator,” Nat. Phys. 7(3), 39. A. Sit et al., “High-dimensional intracity quantum cryptography 219–222 (2011). with structured photons,” Optica 4(9), 1006 (2017). 13. T. Herr et al., “Temporal solitons in optical microresonators,” 40. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. Nat. Photonics 8(2), 145–152 (2014). 13(11), 1900140 (2019). 14. P. K. Maroju et al., “Attosecond pulse shaping using a seeded 41. Y. F. Chen et al., “Laser transverse modes of spherical resonators: free-electron laser,” Nature 578(7795), 386–391 (2020). a review [Invited],” Chin. Opt. Lett. 18(9), 91404 (2020). 15. S. P. Obenschain et al., “Laser-target interaction with induced 42. A. Forbes, M. D. Oliveira, and M. R. Dennis, “Structured light,” spatial incoherence,” Phys. Rev. Lett. 56(26), 2807–2810 (1986). Nat. Photonics 15, 253–262 (2021). 16. A. M. Yao and M. J. Padgett, “Orbital angular momentum: 43. M. V. Berry and M. R. Dennis, “Roadmap on structured light,” origins, behavior and applications,” Adv. Opt. Photonics 3(2), J. Opt. 2(19), 013001 (2017). 161 (2011). 44. O. V. Angelsky et al., “Structured light: ideas and concepts,” 17. Y. Shen et al., “Creation and control of high-dimensional multi- Front. Phys. 8, 114 (2020). partite classically entangled light,” Light Sci Appl 10(1), 50 45. J. Wang et al., “Generation and detection of structured light,” (2021). Front. Phys. 9, 688284 (2021). 18. C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of 46. C. D. E. Otte, “Optical trapping gets structure: structured light for complex vector light fields and their applications,” J. Opt. advanced optical manipulation,” Appl. Phys. Rev. 7(4), 41308 20(12), 123001 (2018). (2020). Advanced Photonics Nexus 024001-18 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 47. Y. Yang et al., “Optical trapping with structured light,” Adv. 74. K. Staliunas and C. O. Weiss, “Tilted and standing waves and Photonics 3(3), 34001 (2021). vortex lattices in class-A lasers,” Physica D 81(1–2), 79–93 48. M. A. Cox et al., “Structured light in turbulence,” IEEE J. Sel. (1995). Top. Quantum Electron. 2(27), 1–21 (2020). 75. J. Swift and P. C. Hohenberg, “Hydrodynamic fluctuations at 49. L. Feng et al. “A review of tunable orbital angular momentum the convective instability,” Phys. Rev. A 15(1), 319–328 (1977). modes in fiber: principle and generation,” Appl. Sci. 9(12), 2408 76. K. Staliunas and C. O. Weiss, “Nonstationary vortex lattices in (2019). large-aperture class B lasers,” J. Opt. Soc. Am. B 12(6), 1142 50. Y. Shen et al., “Optical vortices 30 years on: OAM manipulation (1995). from topological charge to multiple singularities,” Light Sci. 77. G. Huyet et al., “Spatiotemporal dynamics of lasers with a large Appl. 8, 90 (2019). Fresnel number,” Phys. Rev. Lett. 75(22), 4027–4030 (1995). 51. J. Ni et al, “Multidimensional phase singularities in nanophoton- 78. Y. Kuramoto and T. Yamada, “Pattern formation in oscillatory ics,” Science 374, 418 (2021). chemical reactions,” Prog. Theor. Phys. 55, 356 (1976). 52. C. He, Y. Shen, and A. Forbes, “Towards higher-dimensional 79. R. Lefever et al., “Phase dynamics of transverse diffraction structured light,” Light Sci. Appl. 11, 205 (2022). patterns in the laser,” Phys. Lett. A 135(4–5), 254–258 (1989). 53. A. H. Dorrah and F. Capasso, “Tunable structured light with flat 80. N. A. Kudryashov, “Exact solutions of the generalized Kuramoto– optics,” Science 376, 367 (2022). Sivashinsky equation,” Phys. Lett. A 147(5–6), 287–291 (1990). 54. W. T. Buono and A. Forbes, “Nonlinear optics with structured 81. B. J. Gluckman et al., “Time averaging of chaotic spatiotemporal light,” Opto-Electron Adv. 5, 210174 (2022). wave patterns,” Phys. Rev. Lett. 71(13), 2034 (1993). 55. N. G. Basov, V. N. Morozov, and A. N. Oraevskiˇı, “Nonlinear 82. L. A. Lugiato, “Nonlinear optical structures, patterns and chaos,” mode interaction in lasers,” Sov. J. Exp. Theor. Phys. 22, Chaos Soliton. Fract. 4(8/9), 1251–1258 (1994). 895–904 (1966). 83. P. Mandel and M. Tlidi, “Transverse dynamics in cavity non- 56. A. F. Suchkov, “Effect of inhomogeneities on the operation linear optics,” J. Opt. B 6(9), R60 (2004). regime of solid-state lasers,” Sov. J. Exp. Theor. Phys. 49, 84. Y. F. Chen and Y. P. Lan, “Formation of optical vortex lattices in 1495–1903 (1966). solid-state microchip lasers: spontaneous transverse mode lock- 57. R. Graham and H. Haken, “Laserlight–first example of a second- ing,” Phys. Rev. A 64(6), 63807 (2001). order phase transition far away from thermal equilibrium,” 85. Y. F. Chen and Y. P. Lan, “Transverse pattern formation of optical Ztschr. Phys. 237(1), 31–46 (1970). vortices in a microchip laser with a large Fresnel number,” Phys. 58. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimen- Rev. A 65(1), 13802 (2001). sional self-focusing and one-dimensional self-modulation of 86. G. D’Alessandro et al., “Average patterns and coherent phenom- waves in nonlinear media,” J. Exp. Theor. Phys. 34,62–69 ena in wide aperture lasers,” Phys. Rev. E 69(6 Pt 2), 66212 (1972). (2004). 59. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic 87. J. Scheuer and M. Orenstein, “Optical vortices crystals: sponta- diffraction catastrophe,” Philos. Trans. R. Soc. A 291, 453–484 neous generation in nonlinear semiconductor microcavities,” (1979). Science 285(5425), 230–233 (1999). 60. K. I. Kubodera and K. Otsuka, “Single-transverse-mode 88. K. Otsuka and S. Chu, “Generation of vortex array beams from a LiNdP O slab waveguide laser,” J. Appl. Phys. 50(2), 653–659 thin-slice solid-state laser with shaped wide-aperture laser-diode 4 12 (1979). pumping,” Opt. Lett. 34(1), 10–12 (2009). 61. N. B. Baranova et al., “Dislocations of the wavefront of a 89. Y. Shen et al., “Vortex lattices with transverse-mode-locking speckle-inhomogeneous field (theory and experiment),” JETP states switching in a large-aperture off-axis-pumped solid-state Lett. 33, 195 (1981). laser,” J. Opt. Soc. Am. B 35(12), 2940 (2018). 62. L. A. Lugiato, L. M. Narducci, and C. Oldano, “Cooperative 90. L. Allen et al., “Orbital angular momentum of light and trans- frequency locking and stationary spatial structures in lasers,” formation of Laguerre Gaussian laser modes,” Phys. Rev. A J. Opt. Soc. Am. B 5(5), 879 (1988). 45(11), 8185–8189 (1992). 63. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. 91. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian modes Commun. 73(5), 403–408 (1989). of the paraxial wave equation and stable resonators,” J. Opt. Soc. 64. J. V. Moloney and A. C. Newell, “Nonlinear optics,” Physica D Am. A: 21(5), 873–880 (2004). 44(1), 1–37 (1990). 92. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free 65. L. A. Lugiato et al., “Instabilities and spatial complexity in beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). a laser,” J. Opt. Soc. Am. B 7(6), 1019–1033 (1990). 93. V. Garcés-Chávez et al., “Observation of the transfer of the local 66. Q. S. Yang et al, “Global stability and oscillation properties of angular momentum density of a multiringed light beam to an a two-level model for a class-B laser with feedback,” Opt. optically trapped particle,” Phys. Rev. Lett. 91(9), 93602 (2003). Commun. 138(4–6), 325–329 (1997). 94. R. P. Macdonald et al. “Interboard optical data distribution by 67. G. L. Oppo, G. D Alessandro, and W. J. Firth, “Spatiotemporal Bessel beam shadowing,” Opt. Commun. 122(4–6), 169–177 instabilities of lasers in models reduced via center manifold (1996). techniques,” Phys. Rev. A 44(7), 4712–4720 (1991). 95. Y. Li et al., “Classically-entangled Ince–Gaussian modes,” Appl. 68. F. T. Arecchi et al., “Deterministic chaos in laser with injected Phys. Lett. 22(116), 221105 (2020). signal,” Opt. Commun. 51(5), 308–314 (1984). 96. J. C. Gutierrez-Vega and M. A. Bandres, “Helmholtz–Gauss 69. W. Greiner, “Classical electrodynamics,” Phys. Today 52(10), 78 waves,” J. Opt. Soc. Am. A: 22(2), 289–298 (2005). (1999). 97. P. W. Smith, “Simultaneous phase-locking of longitudinal and 70. C. Coste, “Nonlinear Schrödinger equation and superfluid hydro- transverse laser modes,” Appl. Phys. Lett. 13(7), 235–237 (1968). dynamics,” Eur. Phys. J. B 2 245–253 (1998). 98. D. H. Auston, “Forced and spontaneous phase locking of the 71. E. A. Arecchi, “Pattern formation and competition in nonlinear transverse modes of a He–Ne laser,” IEEE J. Quantum Electron. optics,” Phys. Rep. 318(1-6), 1–83 (1999). 4(7), 471–473 (1968). 72. Y. Castin and K. Mo Lmer, “Maxwell–Bloch equations: a unified 99. Z. Zhang and C. Zhao, “Spontaneous phase and frequency lock- view of nonlinear optics and nonlinear atom optics,” Phys. Rev. A ing of transverse modes in different orders,” Phys. Rev. Appl 51(5), R3426 (1995). 13(2), 24010 (2020). 73. K. Staliunas, “Laser Ginzburg–Landau equation and laser hydro- 100. S. Mahler et al., “Improved phase locking of laser arrays with dynamics,” Phys. Rev. A 48(2), 1573 (1993). nonlinear coupling,” Phys. Rev. Lett. 124, 133901 (2020). Advanced Photonics Nexus 024001-19 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 101. A. N. K. Reddy et al., “Phase locking of lasers with Gaussian 128. Y. Li et al., “Sum frequency generation with two orbital angular coupling,” Opt. Express 30(2), 1114–1129 (2022). momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407 102. M. Brambilla et al, “Transverse laser patterns. I. Phase singular- (2015). ity crystals,” Phys. Rev. A 43(9), 5090 (1991). 129. Y. Li et al., “Dynamic mode evolution and phase transition of 103. L. G. Wright, D. N. Christodoulides, and F. W. Wise, twisted light in nonlinear process,” J. Mod. Opt. 63(21), “Spatiotemporal mode-locking in multimode fiber lasers,” 2271–2278 (2016). Science 358(6359), 94–97 (2017). 130. D. G. Pires et al., “Mixing Ince–Gaussian modes through 104. H. Qin et al., “Observation of soliton molecules in a spatiotem- sum-frequency generation,” J. Opt. Soc. Am. B 37(10), 2815 poral mode-locked multimode fiber laser,” Opt. Lett. 43(9), (2020). 1982–1985 (2018). 131. K. Dholakia et al., “Second-harmonic generation and the orbital 105. Y. Ding et al., “Multiple-soliton in spatiotemporal mode-locked angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 multimode fiber lasers,” Opt. Express 27(8), 11435 (2019). (1996). 106. T. Mayteevarunyoo, B. A. Malomed, and D. V. Skryabin, 132. J. Courtial et al., “Second-harmonic generation and the conser- “Spatiotemporal dissipative solitons and vortices in a multi- vation of orbital angular momentum with high-order Laguerre– transverse-mode fiber laser,” Opt. Express 27(26), 37364 (2019). Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997). 107. L. G. Wright et al., “Mechanisms of spatiotemporal mode-lock- 133. H. Yu et al., “Generation of crystal-structure transverse patterns ing,” Nat. Phys. 16(5), 565–570 (2020). via a self-frequency-doubling laser,” Sci. Rep. 3, 1085 (2013). 108. M. Bagci and J. N. Kutz, “Spatiotemporal mode locking in 134. S. Li et al., “Managing orbital angular momentum in second- quadratic nonlinear media,” Phys. Rev. E 102(2-1), 22205 (2020). harmonic generation,” Phys. Rev. A 88(3), 035801 (2013). 109. X. Lin et al., “All few-mode fiber spatiotemporal mode-locked 135. W. T. Buono et al., “Arbitrary orbital angular momentum addition figure-eight laser,” J. Lightwave Technol. 39(17), 5611–5616 in second harmonic generation,” New J. Phys. 16(9), 93041 (2021). (2014). 110. Y. Ding et al., “Spatiotemporal mode-locking in lasers with large 136. Z. Zhou et al., “Orbital angular momentum light frequency modal dispersion,” Phys. Rev. Lett. 126(9), 93901 (2021). conversion and interference with quasi-phase matching crystals,” 111. F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femto- Opt. Express 22(17), 20298 (2014). second pulse shaping and its effects on coherent control,” 137. L. J. Pereira et al., “Orbital-angular-momentum mixing in type-II J. Chem. Phys. 130(3), 34302 (2009). second-harmonic generation,” Phys. Rev. A 96(5), 053856 (2017). 112. G. Pariente et al., “Space-time characterization of ultra-intense 138. Y. Tang et al., “Harmonic spin-orbit angular momentum cascade femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016). in nonlinear optical crystals,” Nat. Photonics 14(11), 658–662 113. Y. Wang et al., “High-power mode-locked 2 μm multimode fiber (2020). laser,” Laser Phys. Lett. 15(8), 85101 (2018). 139. D. G. Pires et al., “Optical mode conversion through nonlinear 114. M. Kowalczyk et al., “Ultrabroadband wavelength-swept source two-wave mixing,” Phys. Rev. A 100(4) (2019). based on total mode-locking of an Yb∶CaF laser,” Photonics 140. D. G. Pires et al., “Higher radial orders of Laguerre–Gaussian Res. 7(2), 182 (2019). beams in nonlinear wave mixing processes,” J. Opt. Soc. Am. 115. K. Krupa et al., “Multimode nonlinear fiber optics, a spatiotem- B 37(5), 1328 (2020). poral avenue,” APL Photonics 4(11), 110901 (2019). 141. D. S. Ding et al., “Linear up-conversion of orbital angular 116. W. H. Renninger, A. Chong, and F. W. Wise, “Pulse shaping and momentum,” Opt. Lett. 37(15), 3270–3272 (2012). evolution in normal-dispersion mode-locked fiber lasers,” IEEE 142. G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral J. Sel. Top. Quantum. Electron. 18, 389–398 (2012). orbital angular momentum transfer via four-wave mixing in 117. A. Chong, L. G. Wright, and F. W. Wise, “Ultrafast fiber lasers Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012). based on self-similar pulse evolution: a review of current 143. H. Yang et al., “Parametric upconversion of Ince–Gaussian progress,” Rep. Prog. Phys. 78, 113901 (2015). modes,” Opt. Lett. 45(11), 3034–3037 (2020). 118. A. Chong et al., “Generation of spatiotemporal optical vortices 144. X. Y. Z. Xiong et al., “Mixing of spin and orbital angular with controllable transverse orbital angular momentum,” Nat. momenta via second-harmonic generation in plasmonic and Photonics 14(6), 350–354 (2020). dielectric chiral nanostructures,” Phys. Rev. B 95(16), 165432 119. A. Mirando et al., “Generation of spatiotemporal optical vortices (2017). with partial temporal coherence,” Opt. Express 29(19), 30426 145. H. Wu et al., “Radial modal transitions of Laguerre–Gauss modes (2021). during parametric up-conversion: towards the full-field selection 120. Q. Cao et al., “Sculpturing spatiotemporal wavepackets with rule of spatial modes,” Phys. Rev. A 101(6), 063805 (2020). chirped pulses,” Photonics Res. 9(11), 2261 (2021). 146. H. Sroor et al., “High-purity orbital angular momentum states 121. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space- from a visible metasurface laser,” Nat. Photonics 14(8), 498– time light sheets,” Nat. Photonics 11(11), 733–740 (2017). 503 (2020). 122. M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal Bessel 147. T. Bell, M. Kgomo, and S. Ngcobo, “Digital laser for on-demand beams: theory and experiments,” Opt. Express 17(20), 18148– intracavity selective excitation of second harmonic higher-order 18164 (2009). modes,” Opt. Express 28(11), 16907 (2020). 123. B. Sun et al., “Four-dimensional light shaping: manipulating 148. A. Srinivasa Rao, K. Miamoto, and T. Omatsu, “Ultraviolet intra- 3þ ultrafast spatiotemporal foci in space and time,” Light Sci. cavity frequency-doubled Pr ∶LiYF orbital Poincaré laser,” Appl. 7(1), 17117 (2018). Opt. Express 28(25), 37397 (2020). 124. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed 149. Z. Zhang et al., “Second harmonic generation of laser beams in optical beams via space-time correlations,” Opt. Express 24(25), transverse mode locking states,” Adv. Photonics 2(4), 026002 28659 (2016). (2022). 125. M. Mounaix et al., “Time reversed optical waves by arbitrary 150. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of vector spatiotemporal field generation,” Nat. Commun. 11, 5813 equilibrium,” Rev. Mod. Phys. 65(3), 851–1112 (1993). (2020). 151. M. Saffman, “Optical pattern formation,” Adv. At. Mol. Opt. 126. C. Wan et al., “Toroidal vortices of light,” Nat. Photonics Phys. 40(8), 229–306 (1998). 16, 519–522 (2022). 152. J. Joly, G. Metivier, and J. Rauch, “Transparent nonlinear 127. A. Beržanskis et al., “Sum-frequency mixing of optical vortices geometric optics and Maxwell–Bloch equations,” J. Differ. Equ. in nonlinear crystals,” Opt. Commun. 150(1), 372–380 (1998). 166, 175–250 (2000). Advanced Photonics Nexus 024001-20 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 153. Y. F. Chen and Y. P. Lan, “Dynamics of the Laguerre Gaussian 179. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for TEM mode in a solid-state laser,” Phys. Rev. A 6(63), 63807 light beams containing orbital angular momentum,” Opt. Lett. 0;l (2001). 24(7), 430–432 (1999). 180. E. Abramochkin and V. Volostnikov, “Beam transformations 154. E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141 (1963). and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 155. N. B. Abraham and W. J. Firth, “Overview of transverse effects in (1991). nonlinear-optical systems,” J. Opt. Soc. Am. B 7(6), 951 (1990). 181. M. W. Beijersbergen et al., “Astigmatic laser mode converters 156. P. Coullet, C. Riera, and C. Tresser, “Stable static localized and transfer of orbital angular momentum,” Opt. Commun. structures in one dimension,” Phys. Rev. Lett. 84(14), 3069 (2000). 96(1–3), 123–132 (1993). 157. K. I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton 182. E. G. Abramochkin and V. G. Volostnikov, “Generalized solutions of the one-dimensional complex Swift–Hohenberg Gaussian beams,” J. Opt. A 6(5), S157 (2004). equation,” Physica D 176(1), 44–66 (2003). 183. E. G. Abramochkin and V. G. Volostnikov, “Generalized 158. B. J. Tepper, “Consequences of abusive supervision,” Acad. Hermite–Laguerre–Gauss beams,” Phys. Wave Phenom. 18(1), Manage. J. 43(2), 178–190 (2000). 14–22 (2010). 159. Y. F. Chen and Y. P. Lan, “Dynamics of helical-wave emission in 184. Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction a fiber-coupled diode end-pumped solid-state laser,” Appl. Phys. lattices and orbital-angular-momentum detection of vortex SU(2) B 73(1), 11–14 (2001). geometric modes,” Opt. Express 26, 25545 (2018). 160. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, 185. Y. Shen et al., “Structured ray-wave vector vortex beams in “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), multiple degrees of freedom from a laser,” Optica 7(7), 820 44 (2004). (2020). 161. B. Zhao et al., “Parabolic-accelerating vector waves,” 186. M. Woerdemann, C. Alpmann, and C. Denz, “Optical assembly Nanophotonics 11(4), 681–688 (2022). of microparticles into highly ordered structures using Ince– 162. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. Gaussian beams,” Appl. Phys. Lett. 98(11), 111101 (2011). 5(10), 1550–1567 (1966). 187. J. A. Davis et al., “Generation of helical Ince–Gaussian beams: 163. Z. Wan et al., “Quadrant-separable multi-singularity vortices beam-shaping with a liquid crystal display,” Proc. SPIE 6290, manipulation by coherent superposed mode with spatial-energy 62900R (2006). mismatch,” Opt. Express 26(26), 34940 (2018). 188. J. B. Bentley et al., “Generation of helical Ince–Gaussian beams 164. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006). beams,” Opt. Lett. 29(2), 144 (2004). 189. S. M. Baumann et al., “Propagation dynamics of optical vortices 165. S. C. Chu and K. Otsuka, “Numerical study for selective exci- due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009). tation of Ince–Gaussian modes in end-pumped solid-state lasers,” 190. E. Louvergneaux et al., “Coupled longitudinal and transverse Opt. Express 15(25), 16506–16519 (2008). self-organization in lasers induced by transverse-mode locking,” 166. N. Barré, M. Romanelli, and M. Brunel, “Role of cavity degen- Phys. Rev. A 57(6), 4899–4904 (1998). eracy for high-order mode excitation in end-pumped solid-state 191. Y. F. Chen, K. F. Huang, and Y. P. Lan, “Spontaneous transverse lasers,” Opt. Lett. 39(4), 1022–1025 (2014). patterns in a microchip laser with a frequency-degenerate reso- 167. D. Mcgloin and K. Dholakia, “Bessel beams: diffraction in a new nator,” Opt. Lett. 28(19), 1811–1813 (2003). light,” Contemp. Phys. 46(1), 15–28 (2005). 192. A. S. Rao et al., “Optical vortex lattice mode generation from a 3þ 168. V. Garcés-Chávez et al., “Simultaneous micromanipulation in diode-pumped Pr ∶LiYF laser,” J. Opt. 23(7), 75502 (2021). multiple planes using a self-reconstructing light beam,” Nature 193. Y. Cao et al., “Transverse patterns and dual-frequency lasing in a 419(6903), 145–147 (2002). low-noise nonplanar-ring orbital-angular-momentum oscillator,” 169. J. C. Gutiérrez-Vega et al., “Alternative formulation for invariant Phys. Rev. Appl. 13(2), 024067 (2020). optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 194. X. Wang et al., “Investigation on the formation of laser transverse (2000). pattern possessing optical lattices,” Front. Phys. 9(1), 801916 170. J. C. Gutiérrez-Vega et al., “Experimental demonstration of (2022). optical Mathieu beams,” Opt. Commun. 195(1–4), 35–40 (2001). 195. E. Louvergneaux et al., “Transverse mode competition in a CO 171. S. Chávez-Cerda et al., “Holographic generation and orbital laser,” Phys. Rev. A 53(6), 4435 (1996). angular momentum of high-order Mathieu beams,” J. Opt. Soc. 196. R. Oron et al., “Laser mode discrimination with intra-cavity Am. B 4(2), S52–S57 (2002). spiral phase elements,” Opt. Commun. 169(1–6), 115–121 (1999). 172. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite 197. K. Sueda et al., “Laguerre–Gaussian beam generated with a energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). multilevel spiral phase plate for high intensity laser pulses,” 173. G. A. Siviloglou et al., “Observation of accelerating Airy beams,” Opt. Express 12(15), 3548–3553 (2004). Phys. Rev. Lett. 4, S52 (2007). 198. W. M. Lee, X. C. Yuan, and W. C. Cheong, “Optical vortex beam 174. Y. Shen et al., “SU(2) Poincaré sphere: a generalized represen- shaping by use of highly efficient irregular spiral phase plates for tation for multidimensional structured light,” Phys. Rev. A optical micromanipulation,” Opt. Lett. 29(15), 1796–1798 (2004). 102(3), 031501 (2020). 199. D. J. Kim and J. W. Kim, “High-power TEM00 and Laguerre– 175. E. Abramochkin and T. Alieva, “Closed-form expression for Gaussian mode generation in double resonator configuration,” mutual intensity evolution of Hermite–Laguerre–Gaussian Schell- Appl. Phys. B 121(3), 401–405 (2015). model beams,” Opt. Lett. 42(19), 4032 (2017). 200. D. J. Kim, J. I. Mackenzie, and J. W. Kim, “Adaptable beam 176. W. N. Plick et al., “Quantum orbital angular momentum of profiles from a dual-cavity Nd:YAG laser,” Opt. Lett. 41(8), elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013). 1740–1743 (2016). 177. Y. Shen et al., “Periodic-trajectory-controlled, coherent-state- 201. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital phase-switched, and wavelength-tunable SU(2) geometric modes angular momentum conversion in inhomogeneous anisotropic in a frequency-degenerate resonator,” Appl. Opt. 57(32), 9543 media,” Phys. Rev. Lett. 96(16), 163905 (2006). (2018). 202. L. Marrucci, “The q-plate and its future,” J. Nanophotonics 7(1), 178. Y. Shen et al., “Hybrid topological evolution of multi-singularity 078598 (2013). vortex beams: generalized nature for helical-Ince–Gaussian and 203. B. Piccirillo et al., “Photon spin-to-orbital angular momentum Hermite–Laguerre-Gaussian modes,” J. Opt. Soc. Am. A: 36, 578 conversion via an electrically tunable q-plate,” Appl. Phys. Lett. (2019). 97(24), 241104 (2010). Advanced Photonics Nexus 024001-21 Mar∕Apr 2023 Vol. 2(2) Wang et al.: Evolution on spatial patterns of structured laser beams: from spontaneous organization… 204. D. Naidoo et al., “Controlled generation of higher-order Poincaré 210. C. W. Johnson et al., “Exact design of complex amplitude holo- sphere beams from a laser,” Nat. Photonics 10(5), 327–332 grams for producing arbitrary scalar fields,” Opt. Express 28(12), (2016). 17334–17346 (2020). 205. R. R. Alfano, G. Milione, E. J. Galvez, and L. Shi, “A laser for 211. C. Rosales-Guzmán et al., “Polarisation-insensitive generation of complex spatial modes,” Nat. Photonics 10(5), 286–288 (2016). complex vector modes from a digital micromirror device,” Sci. 206. S. Ngcobo et al., “A digital laser for on-demand laser modes,” Rep. 10(1), 10434 (2020). Nat. Commun. 4, 2289 (2013). 212. P. Chen et al., “Liquid-crystal-mediated geometric phase: from 207. L. Burger et al., “Implementation of a spatial light modulator for transmissive to broadband reflective planar optics,” Adv. Mater. intracavity beam shaping,” J. Opt. 17(1), 15604 (2015). 32(27), 1903665 (2019). 208. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection 213. Y. Shen and C. Rosales-Guzmán, “Nonseparable states of light: of optical modes with spatial light modulators,” Adv. Opt. from quantum to classical,” Laser Photonics Rev. 16(7), 2100533 Photonics 8(2), 200 (2016). (2022). 209. S. Scholes et al., “Structured light with digital micromirror de- vices: a guide to best practice,” Opt. Eng. 59(4), 041202 (2020). Biographies of the authors are not available. Advanced Photonics Nexus 024001-22 Mar∕Apr 2023 Vol. 2(2)
Journal
Advanced Photonics Nexus – SPIE
Published: Mar 1, 2023
Keywords: spatial patterns; transverse modes; spatiotemporal beams; structured laser beams; nonlinear optics