Fig. 1. Timeline of the evolution on spatial patterns of structured laser beams.

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 nonstationary 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.

Fig. 3. Basic types of transverse patterns. (a) HG beam. (b) LG beam. (c) IG beam. (d) Mathieu beam. (e) Bessel beam. (f) Airy beam. (a)–(f) are adapted from Ref. 42. (g) Parabolic beam. Adapted from Ref. 160. (h) Parabolic-accelerating vector beam. Adapted from Ref. 161.

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 HLG31 modes with different values of α. Adapted from Ref. 175. (d) The intensity distribution of even odd IG mode with p=5 and m=3, and the HIG mode generated by the corresponding superposition when ε=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 (n,m)=(0,6). Adapted from Ref. 178.

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.

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 supporting 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 (TEM00 and TEM01) modes to create scanning beam. Adapted from Ref. 114.

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.

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.

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.