Amplification and generation of ultra-intense twisted laser pulses via stimulated Raman scattering

Twisted Laguerre–Gaussian lasers, with orbital angular momentum and characterized by doughnut-shaped intensity profiles, provide a transformative set of tools and research directions in a growing range of fields and applications, from super-resolution microcopy and ultra-fast optical communications to quantum computing and astrophysics. The impact of twisted light is widening as recent numerical calculations provided solutions to long-standing challenges in plasma-based acceleration by allowing for high-gradient positron acceleration. The production of ultra-high-intensity twisted laser pulses could then also have a broad influence on relativistic laser–matter interactions. Here we show theoretically and with ab initio three-dimensional particle-in-cell simulations that stimulated Raman backscattering can generate and amplify twisted lasers to petawatt intensities in plasmas. This work may open new research directions in nonlinear optics and high–energy-density science, compact plasma-based accelerators and light sources.


SUPPLEMENTARY NOTE 1 -STIMULATED RAMAN BACKSCATTERING
To derive Orbital Angular Momentum (OAM) selection rules for Raman amplification we write pump and seed laser vector potentials as A pump = A 0 (r, φ, t) exp(ik 0 z −iω 0 t)+c.c. and where A 0 (r) and A 1 (r) are arbitrary functions of the transverse coordinate r, being slowly varying envelopes for the pump (A 0 ) and seed (A 1 ) respectively, and where (k 0 , ω 0 ) and (k 1 , ω 1 ) are the pump and seed lasers wavenumber (k) and frequency (ω) respectively. Note that we use k 0 and −k 1 to indicate that the pulses travel in opposite directions. In addition, we write the plasma electron density perturbation (Langmuir wave) as (n e − n 0 )/n 0 = δn exp(ik p z − iω p t) + c.c., where δn is a slowly varying envelope, and where (k p , ω p ) are, respectively, the plasma wavenumber (k p ≈ 2k 0 − ω p /c) and frequency ω p = e 2 n 0 / 0 m e , with c the speed of light, e the elementary charge, n 0 the background plasma density, 0 the vacuum electric permittivity, and m e the electron mass. We use cylindrical coordinates where (r, φ, z) are the radial distance to the axis (r), the azimuthal angle (φ), and the longitudinal distance (z). Using the slowly varying envelope approximation, the following equations describing Raman backscattering can be derived [1][2][3][4]: where the superscript * denotes the complex conjugate and where the operators D 0 , D 1 and D p are given by: where Eq. (6) is strictly valid for cold plasmas. In addition to Eqs. (1)(2)(3)(4)(5)(6), the lasers and plasma wavenumber and frequency matching conditions are given by k 0 = k 1 + k p , ω 0 = ω 1 + ω p . Each laser also obeys the dispersion relation of electromagnetic waves in a plasma, where k 2 0,1 c 2 = ω 2 0,1 − ω 2 p . We can simplify Eq. (1-3) by making a general assumption that the vector potential envelope of each laser can be written as A = A x (t, z)T x (r ⊥ , z)e x + A y (t, z)T y (r ⊥ , z)e y , where A x/y is a function of t and z and represents the longitudinal envelope profile, and where T x/y is a function of the coordinates r ⊥ and z, and represents the transverse envelope profile. We then use A 0,x/y T 0,x/y and A 1,x/y T 1,x/y to designate the pump and seed fields in the transverse x and y directions respectively. We can also assume that plasma density perturbations can be written as δn = δn(t, z)T δn . Although our calculations are valid for any T x,y , for a Laguerre-Gaussian mode with OAM level and radial mode p, T x,y (or T δn ) is given by: where we dropped the subscripts for simplicity, where is the Laguerre-Gaussian polynomial with azimutal index and radial index p, and where is the Gouy phase shift. In addition, z r = kw 2 0 /2 is the Rayleigh length, k is the wavenumber, w 0 is the spot-size at focus, and w(z) as A x and A y are functions of (t, z) only, and since ∂T x/y /∂t = 0 because T x/y depends on (x ⊥ , z) only, inserting Eq. (7) into Eq. (4) and (5) yields: To explore all selection rules we can assume that the transverse laser envelopes T 0 and T 1 obey to the paraxial approximation: Using Eqs. (12) and (13), Eqs. (10) and (11) become given by: Using Eqs. (14) and (15), we can then recast Eqs. (1-2) as: We note that Eqs. (18)  In its most simple configuration considering lasers with identical transverse profiles in every polarisation direction, it is possible to readily recover the usual plane wave solutions.
Cancellation of all phase factors leads to: For Laguerre Gaussian lasers with orbital angular momentum, cancelling phase factors readily implies the conservation of orbital angular momentum, 0 = 1 + p , the conservation of energy, ω 0 = ω 1 + ω p , and the conservation of linear momentum, It is possible to further simplify Eqs. (14) and (15) assuming that the spot-size does not change during propagation (a valid approximation to interpret our simulations since the Rayleigh length is much larger than the interaction length). In this case ∂ z T A = (14) and (15) may also take the more familiar form of the plane wave equations for Raman amplification [3,5]: Hence the 1D spatial-temporal amplification process is independent of the transverse laser profiles, under the condition that the above selection rule for angular momentum conservation is obeyed, in addition to the usual selection rules for conservation of energy and linear momentum.
In order to derive scalings for the growth rate of the instability, we assume a long laser such that k 0,1 ∂A (0,1) /∂z ω 0,1 ∂A (0,1) /∂t, in order to explore the temporal problem (note that this same approximation could be done directly to Eqs. (14) and (15) yielding the same result, without the need of assuming that A ⊥ was a function of z only). In this case, stimulated Raman scattering equations become: The selection rules for the orbital angular momentum, which ensure conservation of angular momentum described in the main Manuscript, also follow directly from Eqns. (23)-(25).
These rules will be derived and studied in more detail below. We note that Eqs. (23)- (25) are valid for arbitrary transverse pump and seed envelope profiles.
We can derive Raman backscattering growth rates assuming that the amplitude of the pump laser remains constant during its interaction with the plasma wave and the seed laser.
The latter assumption also implies that the intensity of the seed pulse is much smaller than the intensity of the pump laser. Thus, combining Eq. (24) with the time derivative of Eq. (25) and neglecting ∂ t A 0 (t), we find: Equation (26) can be further simplified by substituting Eq. (24) into Eq. (26) giving: Equation (27), which can be solved exactly, determines the plasma density perturbation associated with the interaction between pump and seed. By assuming that δn * (t = 0) = 0, 25)), leads to: where Γ is the growth rate of the instability.
The temporal evolution of the seed is found by combining Eq. (28) with Eq. (24): where C is a constant specified by the initial seed laser vector potential profile. Equation (30) describes the temporal growth of the Raman backscattering amplified seed pulse assuming a non-evolving pump laser with an arbitrary transverse envelope.
The derivation of Eq. with the creation and amplification of new OAM modes. We have confirmed all these predictions using 3D PIC simulations. Therefore inclusion of pump depletion effects will not affect the phenomenology for the OAM beam amplification, although they may change the growth rates according to Ref. [6].

SUPPLEMENTARY NOTE 2 -AMPLIFICATION OF AN EXISTING OAM MODE
In this section, we study the Raman amplification of laser pulses with orbital angular momentum, and derive selection rules for the OAM of the pump and seed pulses and the plasma wave. In this and the subsequent sections, we assume that each laser with OAM laser can be described as a laser ∼ exp(i laser φ) and each plasma wave OAM mode can be described by δn ∼ exp(i p φ).
We start by studying the case leading to the amplification of an existing OAM seed. We then consider a seed laser with OAM component given by a 1 ∼ exp (i 1x φ) e x , and a pump with a 0 ∼ exp (i 0x φ) e x where e x is the unit vector in the x direction, indicating the direction of polarisation. Direct substituion in Eq. (25) shows that the pump and the seed create a OAM plasma wave perturbation with δn ∼ exp (i p φ). Thus, the OAM of the plasma wave is p = 0x − 1x , ensuring angular momentum conservation. The same selection rule has also been derived above in Eqs. (23) and (24), proving that these equations are mutually consistent. Thus, a pump with a single, but arbitrary OAM mode, or even without any OAM at all, can be used to amplify a seed pulse with a single and also arbitrary OAM mode, because the plasma wave will carry all excess angular momentum. This is schematically shown in Fig. 1 A similar calculation can be performed for the case of circular polarisation, leading to the same selection rules.

SUPPLEMENTARY NOTE 3 -GENERATION OF NEW OAM MODES
In order to explore the generation and amplification of new OAM modes in the seed, we consider a pump with a 0 ∼ exp(i 0x φ)e x + exp(i 0y φ)e y , and a seed with Hence, the pump has an OAM mode linearly polarised in each transverse direction, and the seed contains a single OAM mode linearly polarised in the x direction. According to  and in y is 0y = 1, and where the initial seed is polarised in x with 1x = 1. In Fig. 2(a), the x pump component interacts with the existing OAM seed component in x to produce a plasma wave with OAM given by p = 0x − 1x = −1 (green dashed line). Figure 2(b) illustrates this second step where the matching conditions are satisfied for the y direction once a new seed is created in that direction. In Fig. 2(b), the plasma wave then interacts with the pump polarised in y and lead to a new seed polarised in y with 1y = 0y − p = 2 (orange dashed line) In addition, the existing seed mode linearly polarised in x [exp(i 1x φ)] will continue to be amplified. Figure 2 illustrates these steps leading to the generation of a new linearly polarised OAM seed. It is important to note that the new 1y mode in the seed cannot interact with the 0x mode in the pump since these modes have orthogonal polarisation.
An identical setup can also be used to generate and amplify a new mode with circular polarisation. This is possible because right-handed and left-handed circularly polarised modes do not interact, just as linearly polarised modes in two orthogonal directions do not interact.
We now consider an initial pump with a 0 ∼ exp(i 0+ φ) (e x + ie y )+exp(i 0− φ) (e x − ie y ) and an initial seed with a 1 ∼ exp(i 1+ φ) (e x + ie y ). Hence, the pump has a single OAM mode 0+ circularly polarised in e + and a single OAM mode 0− circularly polarised in e − . The seed, with an initial single OAM mode 1+ , is circularly polarised in the anti-clock wise direction.
According to Eq. (30), the anti-clock wise seed and pump generate a plasma wave with p = 0+ − 1+ . The plasma wave then interacts with the pump circularly polarised in the clockwise direction and produces a new seed mode, also circularly polarised in the clockwise direction Direct substitution of pump and seed expressions in (e x − ie y ), consistent with the conservation of orbital angular momentum. As a result, and in addition to the amplification of the existing circularly polarised mode, a new circularly polarised OAM seed can also be produced and amplified with a handedness opposite to the initial seed. Figure 3 shows the generation of OAM from initial TEM modes, where we consider a pump given by a 0 ∼ TEM 00 (e x + e y ), and an initial seed given by a 1 ∼ TEM 10 e x + iTEM 01 e y , i.e. the y seed component is π/2 out of phase with respect to the seed x com-ponent. Initial pump modes are shown in Figs. 3(a) and (c). The TEM 00 pump polarised in x then interacts with the TEM 10 seed also polarised in x to generate a TEM plasma wave with TEM 10 (plasma wave represented by the green dashed lines in Fig. 3(a)). This daughter plasma wave then interacts with the TEM 00 pump polarised in y yielding a new TEM 10 seed mode polarised in y (new seed mode in y represented by the orange dashed lines in Fig. 3(b)). Simultaneously, the TEM 00 pump polarised in y interacts with the iTEM 01 seed also polarised in y to generate a plasma wave with iTEM 01 . (new plasma wave mode represented in dashed green in Fig. 3c). This daughter plasma wave then interacts with the TEM 00 pump polarised in x and generates a new iTEM 01 seed polarised in x (new seed in

SUPPLEMENTARY NOTE 4 -CONVERSION FROM TEM TO OAM MODES
x represented by orange dashed lines in Fig. 3d). As a result, a new seed is created and amplified with a 1 ∼ (TEM 10 + iTEM 01 )(e x + e y ). According to Ref. [7], this corresponds to new OAM modes with 1x = 1y = 1. Equivalently, substituting pump and seed expressions into Eq. (30) leads to a 2 ∼ TEM 00 (TEM 10 + iTEM 01 ) (e x + e y ), where TEM 00 ∼ 1 in this context.
An alternative way to view this scheme is to define new ortogonal unit vectors given by e 1,2 = (e x ± e y )/ √ 2 and write the seed pulse as a 1 ∼ (TEM 10 + iTEM 01 )e 1 + (TEM 10 − iTEM 01 )e 2 . While this pulse has no overall OAM, the e 1 and e 2 components have OAM of = 1 and = −1 respectively. Since the pump pulse is given by a 0 ∼ TEM 00 e 1 , it will amplify the = 1 mode (same polarisation) while leaving the = −1 mode untouched (orthogonal polarisation). The amplified seed pulse will then end up with overal OAM of level = 1, while it had no OAM initially. This example also illustrates that it is possible to selectively amplify OAM modes in the seed pulse based on their polarisation relative to the polarisation of the pump pulse.

SUPPLEMENTARY NOTE 5 -PLASMAS AND KERR MEDIA
In stimulated Raman scattering in plasmas the electromagnetic pump wave decays into a scattered electromagnetic wave with the Langmuir wave providing the coupling between the other two electromagnetic waves. In other nonlinear optical media, the role of the Langmuir wave would be replaced by molecular vibrations, for example. Equations for three wave mixing processes in nonlinear anisotropic optical media with Kerr nonlinearity are [8]: where A We now assume that A