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Creating high-quality vector vortex (VV) beams is possible with a myriad of techniques at low power, and while a few studies have produced such beams at high-power, none have considered the impact of amplification on the vector purity. Here we employ novel tools to study the amplification of VV beams, and in particular the purity of such modes. We outline a general toolbox for such investigations and demonstrate its use in the general case of VV beams through a birefringent gain medium. Intriguingly, we show that it is possible to enhance the purity of such beams during amplification, paving the way for high-brightness VV beams, a requirement for their use in high-power applications such as laser materials processing.

Laser brightness is a measure of the ability to de- liver intense light to a target, and encapsulates both the energy content and the beam quality. High brightness lasers requires that both parameters be maximised, yet standard laser cavities do not allow this. For example, in solid-state lasers multimode beams have a high energy content but low beam quality, while Gaussian modes have a small mode volume and hence low energy extraction, but in a good quality mode. Here we over- come this fundamental limitation and demonstrate an optimal approach to realising high brightness lasers. We employ intra- cavity beam shaping to produce a Gaussian mode that carries all the energy of the multimode beam, thus energy extraction and beam quality are simultaneously maximised. This work will have a significant influence on the design of future high brightness laser cavities.

The angular momentum state of light can be described by positions on a higher-order Poincar\'e (HOP) sphere, where superpositions of spin and orbital angular momentum states give rise to laser beams that have found many applications, including optical communication, quantum information processing, microscopy, optical trapping and tweezing and materials processing. Many techniques exist to create such beams but none to date allow their creation at the source. Here we report on a new class of laser that is able to generate all states on the HOP sphere. We exploit geometric phase control with a non-homogenous polarization optic and a wave-plate inside a laser cavity to map spin angular momentum (SAM) to orbital angular momentum (OAM). Rotation of these two elements provides the necessary degrees of freedom to traverse the entire HOP sphere. As a result, we are able to demonstrate that the OAM degeneracy of a standard laser cavity may be broken, producing pure OAM modes as the output, and that generalized vector vortex beams may be created from the same laser, for example, radially and azimuthally polarized laser beams. It is noteworthy that all other aspects of the laser cavity follow a standard design, facilitating easy implementation.

It is well-known how to control the spatial output from a laser, with most solutions to date involving customised intra-cavity elements in the form of apertures, diffractive optics and free-form mirrors. These optical elements require considerable design and fabrication effort and suffer from the further disadvantage of being immutably connected to the selection of a particular spatial mode. Consequently, most laser systems are designed for the ubiquitous Gaussian mode, whereas it is clear that there are many instances when a customised mode would be preferable. We overcome these limitations with the first digital laser, comprising an electrically addressed reflective phase-only spatial light modulator as an intra-cavity holographic mirror. The phase and amplitude of the holographic mirror may be controlled as simply as writing a new gray-scale image (computer generated hologram) to the device: on-demand laser modes. We show that we can digitally control the laser modes with ease, albeit with higher round-trip losses and thus requiring higher gain, and demonstrate the versatility of the technique by switching between several spatial modes in an otherwise standard solid-state laser resonator., Comment: 9 pages