The nonlinear optical properties of the plasma are mostly determined by the motion of background plasma electrons. Their motion is not bound to physical boundaries, and can be shaped nearly arbitrarily, provided that it is subject to appropriately tailored electric and magnetic fields. Hence, in addition to supporting very high intensities without breaking, the plasma can be seen as a flexible optical element, whose properties can be nearly arbitrarily shaped. Ultra-intense vortex laser pulses carrying OAM provide an additional degree of freedom to control the motion of plasma electrons at ultra-high intensities. These vortex beams are then attractive to manipulate the nonlinear optics of plasmas, and to be, in turn, manipulated by the plasmonic structures themselves.

#### 2.2.1. Generation and Amplification of Ultra-Intense Vortex Beams

An attractive path to create ultra-intense vortex laser pulses leverages on Raman amplification [

50], relying on stimulated Raman backscattering [

62,

63,

64]. Just as in parametric amplification using a nonlinear crystal, Raman amplification relies on a three wave process to transfer the energy of a long pump laser pulse, with frequency

${\omega}_{0}$ and wavenumber

${k}_{0}$, to a counter propagating seed laser pulse, with

$({\omega}_{1},{k}_{1})$, through the generation of a plasma wave during the interaction, with

$({\omega}_{p},{k}_{p})$. The plasma wave plays the role of an idler wave in parametric amplification.

Stimulated Raman backscattering occurs when the frequency and wavenumber of the intervening lasers satisfy the matching conditions that ensure photon energy and momentum conservation. In the absence of OAM, seed pulse amplification then occurs when

${\omega}_{p}={\omega}_{0}-{\omega}_{1}$. In addition, the corresponding matching condition for the wavenumber is

$({k}_{p}={k}_{0}-{k}_{1})$. The matching conditions for

$\omega $ and

k need to be supplemented by an additional condition in the presence of twisted light. This additional matching condition, which expresses conservation of OAM, leads to the generation of a plasma wave with OAM. Similarly to the wavenumber matching condition, the OAM level of the plasma wave is given

${\ell}_{p}={\ell}_{0}-{\ell}_{1}$, where

$({\ell}_{p},{\ell}_{0},{\ell}_{1})$ are the orbital angular momentum levels of the plasma wave

$({\ell}_{p})$, pump

$({\ell}_{0})$, and seed

$({\ell}_{1})$. Generation (and amplification) of new OAM modes can also occur in Raman amplification when the pump beam has field components in two orthogonal polarisations, with OAM

${\ell}_{0\parallel}$ and

${\ell}_{0\perp}$ in each polarisation [

50,

65]. Hence, if the seed is initially polarised only along one of them, with

${\ell}_{1\parallel}$, a new mode, with

${\ell}_{1\perp}={\ell}_{1\parallel}+{\ell}_{0\parallel}-{\ell}_{0\perp}$ appears to ensure momentum conservation.

Because plasma electrons can usually adjust to the beating structure of the lasers, the plasma wave can always absorb the momentum mismatch (both the linear momentum, associated with the axial laser wavenumber, and the OAM) between seed and pump lasers. Thus, the amplification of a twisted laser pulse is possible even when the pump laser contains no OAM. This property is very attractive because the amplification of twisted light and non-twisted light can then be achieved using identical experimental setups.

#### 2.2.2. High Orbital Angular Momentum Harmonics in Plasmas

Because of energy conservation, the

nth harmonic of a laser with frequency

$\omega $ and orbital angular momentum

ℓ is characterised by frequency

$n\omega $ and orbital angular momentum

$n\ell $. Apart from an exceptional experimental result where this has not been observed [

16], conservation of energy and angular momentum is usually expected to occur, and has been demonstrated experimentally [

19,

66]. These fundamental conservation laws appear to suggest that the OAM is always coupled to the high frequency harmonics. In strong contrast with these observations, Raman amplification can be used to generate high OAM harmonics while keeping the laser frequency unchanged, thereby manipulating the OAM as an independent degree of freedom. As a result, it may open the way to produce ultra-short vortex beams over the entire frequency spectrum and OAM range if combined with some other high harmonic generation mechanism [

51].

The generation of OAM harmonics in Raman amplification can be achieved by considering a pump beam consisting of a superposition of modes with different OAM modes, as shown in

Figure 3a. The high OAM harmonics obey a simple algebraic rule, where the OAM of the

nth harmonic is given by

${\ell}_{n}={\ell}_{1}\pm n\Delta \ell $, where

n is an integer that refers to the harmonic order and

$\Delta \ell ={\ell}_{01}-{\ell}_{00}$ is the difference between the OAM levels in the pump. All harmonics then appear when

$\Delta \ell =1$, even harmonics are created when both

$\Delta \ell $ and

${\ell}_{1}$ are even, and odd when

$\Delta \ell $ is even and

${\ell}_{1}$ is odd.

The high OAM harmonics are created through an angular momentum cascade, where the energy flows from lower OAM components to neighbouring modes with higher OAM. An initial seed with

${\ell}_{1}$ and pump with

${\ell}_{00}$ and

${\ell}_{01}$, create a plasma wave that is a combination of two OAM modes, with

${\ell}_{00}-{\ell}_{1}$ and

${\ell}_{01}-{\ell}_{1}$. The

${\ell}_{00}$ pump mode beats with the plasma wave

${\ell}_{01}-{\ell}_{1}$ produces a sideband component in the seed pulse with

${\ell}_{1}+\Delta \ell $ (

$\Delta \ell \equiv {\ell}_{00}-{\ell}_{01}$). The other pump mode with

${\ell}_{01}$ beats with the remaining plasma wave OAM component, with

${\ell}_{00}-{\ell}_{1}$, creating a seed sideband component with

${\ell}_{1}-\Delta \ell $. Each of these seed sidebands will continue to beat with the pump, adding new OAM harmonics. The high harmonics will be generated at least up to the order allowed by the paraxial approximation, valid as long as the azimutal laser wavenumber (associated with the OAM) is smaller than the longitudinal laser wavenumber (red line in

Figure 3b). If combined with conventional high harmonics generation, where the frequency and the OAM harmonics are upshifted by the same factor (red line in

Figure 3b), the scheme may enable the generation of light over the full OAM-frequency spectrum (shaded blue in

Figure 3b).

The generation of the OAM harmonics usually requires orbital angular momentum to be initially present. This requirement can be relaxed if circularly polarised lasers are used instead. According to theoretical and numerical calculations, the plasma can be used to convert SAM into OAM in second harmonic generation of circularly polarised photons [

52]. Being an illustration of energy and angular momentum conservation, this process is also a demonstration of the spin to orbital angular momentum conversion. The process occurs when a circularly polarised photon, with wavenumber

${k}_{0}$ and spin angular momentum

$s=1$, interacts with a plasmon, with wavenumber

${k}_{p}$ and with no angular momentum. The plasmon can be initially produced by any high order process. During the interaction, a virtual plasmon appears with a frequency close to the optical frequency, with a wavenumber

${k}_{0}+{k}_{p}$, and with an angular momentum associated with the initial photon spin. Because of energy and momentum conservation, the interaction between the virtual plasmon with another spin polarised photon produces a second harmonic photon with wavenumber

$2{k}_{0}+{k}_{p}$ and angular momentum

$s+\ell =2$. Because

s cannot be larger than the unity, the upshifted photon OAM is finite and given by

$\ell =1$.

High vortex harmonic generation can also occur as an intense laser pulse propagates in an underdense plasma. This configuration was investigated theoretically and with numerical simulations. The mechanism for the generation of the high vortex harmonics depends on the laser polarisation. In linear polarisation, the high harmonics appear as a result of a cascade involving four-wave mixing. In circular polarisation, a three-wave mixing cascade is involved [

53].

High harmonic generation using vortex beams has also been investigated theoretically, numerically and experimentally, in a configuration where an ultra-intense twisted laser pulse interacts with a plasma mirror, a solid density target where

${\omega}_{0}<{\omega}_{p}$. The high harmonic generation model that is usually employed at laser intensities above

$I{\lambda}^{2}>1.37\times {10}^{18}\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}\mathsf{\mu}{\mathrm{m}}^{2}$ is the relativistic oscillating mirror (ROM) [

54]. At these intensities, the quiver motion of the electrons at the target surface is relativistic. As electrons execute periodic oscillations in the laser fields, they also emit radiation at very high harmonics, which can reach the XUV for IR lasers with central wavelength

${\lambda}_{0}\sim 1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m. In the presence of twisted light, the oscillations at the target surface have an azimutal phase dependence, forming a vortex oscillating mirror (VOM) [

55]. The reflected light electric (or magnetic) field, given by

$E\sim {a}_{0}sin\left\{{\omega}_{0}t+\ell \varphi +\alpha sin\left[2({\omega}_{0}t+\ell \varphi )\right]\right\}$, can be decomposed in a Fourier series as

$E/{a}_{0}\sim {\Sigma}_{n=0}^{\infty}{J}_{n}(\alpha )sin\left[(2n+1)({\omega}_{0}t+\ell \varphi )\right]$, where

${a}_{0}$ is the peak normalised laser vector potential, related to the laser intensity through

${a}_{0}\simeq 8.6\times {10}^{-10}\lambda [\mathsf{\mu}$m

$]{I}^{1/2}[\mathrm{W}/{\mathrm{cm}}^{2}]$ and

$\alpha $ is a parameter related to the amplitude of the electron oscillations. The highest harmonic order is set by the plasma frequency,

${\omega}_{p}$, of the solid density. Thus, efficient generation of photons with

$n\omega >{\omega}_{p}$ occurs. In addition, and just as in the ROM, only the odd harmonics appear.

The coherent wake emission (CWE) is a distinct mechanism for high harmonics generation, which operates at intensities well below the relativistic regime, for

$I{\lambda}^{2}>1.37\times {10}^{18}\phantom{\rule{3.33333pt}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}\mathsf{\mu}{\mathrm{m}}^{2}$ [

56]. In the CWE, the high harmonics are generated by the periodic motion of electron bunches which are generated near the critical density. These electron bunches oscillate across the overdense plasma region, emitting a train of attosecond pulses that can be decomposed in high frequency harmonics that can extend up to the extreme ultraviolet.

The CWE of vortex light has been demonstrated experimentally, employing the 100 TW-class Ti:sapphire laser facility at CEA-IRAMIS, delivering

$25\phantom{\rule{3.33333pt}{0ex}}\mathrm{fs}$ laser pulses [

57]. The optical vortex of the main laser pulse, used to drive the high harmonics, was created experimentally by introducing a spiral phase plate in the collimated laser pulse. After the spiral phase plate, the laser acquired an OAM level

$\ell =1$. The OAM corresponding to each frequency harmonic was retrieved with the aid of the interference pattern between the interacting high power beam and a reference Gaussian laser pulse. The reconstructed phase profile corresponding to each harmonic provided a clear demonstration that the upshifted frequency photons with frequency

$n\omega $ were also characterised by an OAM

$n\ell $.

#### 2.2.3. Plasma Holograms and Phase Plates for Ultra-Intense Lasers

The use of transient plasma gratings [

58] to manipulate light at ultra-high intensities has attracted considerable experimental attention as it can be used to generate high power vortex laser beams, and their harmonics, at ultra-high intensities. Instead of being formed in an underdense plasma, as in Raman scattering, a transient plasma grating can be formed at the surface of a solid target. The use of this concept to generate intense vortex light has been recently demonstrated experimentally [

59].

The starting point to understand how a transient plasma grating can form is the observation that the plasma expansion velocity at the surface of a solid target can be controlled by the local laser fluence. Consider, for instance, the interference pattern of two laser pulses colliding at an angle, characterised by a sinusoidal fluence variation. Focusing these lasers at the surface of a solid target will lead to the generation of a sinusoidally varying density profile, a transient plasma grating. Because the plasma is not bound to physical boundaries, the electron expansion can assume the shape of complex interference patterns. Using a spiral phase plate to create a vortex beam from one of the two interfering lasers, the object beam, leads to a complex interference pattern that results in a forked plasma grating. This profile can be imprinted at the surface of the solid target, forming the plasma hologram. The information stored in the transient grating can be recovered by reflecting an ultra-intense laser pulse onto its surface. This procedure allows to create very high energy replicas of the object beam in the diffracted orders. At intensities above ${10}^{16}$$\mathrm{W}/{\mathrm{cm}}^{2}$ the response of the transient plasma grating becomes strongly nonlinear, leading to the generation of high harmonics.

A complementary theoretical proposal for the generation of ultra-intense vortex beams consists of using a solid density plasma target with a azimuthally varying thickness. Numerical simulations demonstrated that an intense Gaussian beam without orbital angular momentum interacting with this plasma-based spiral phase-plate could be used to generate OAM beams.