The Gaussian Beam [link] is not the only available solution to the Helmholtz equation [link]. The Hermite-Gaussian Beam also satisfies the Helmholtz equation and it shares the same wavefronts (shape) of the Gaussian Beam. Where it differs is in the distribution of intensity in the beam. The Hermite-Gaussian Beam distribution is a modulated Gaussian distribution in the x and y directions which can be seen as a number of functions in superposition. The below figures depict the cross-sections of ascending order intensity distributions for the Hermite-Gaussian Beam. Secondly, distribution orders zero through three are shown.
The Complex amplitude of the Hermite-Gaussian beam labeled by indexes l,m (orders):
The Laguerre-Gaussian Beam is a solution to the Helmholtz equation in cylindrical coordinates.
The shape of the Laguerre-Gaussian Beam intensity distribution is of a toroid which increases in radius for orders where m = 0 and for orders m > 0, it takes the form of multiple rings.
The Bessel Beam
The Bessel Beam, by comparison to the Gaussian Beam differs in that it has a ripple effect by oscillation in addition to a similar gaussian curve. The complex amplitude of the Bessel Beam is an exact solution to the Helmholtz equation, while the complex amplitude of the Gaussian beam is an approximate solution (paraxial solution).
B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.