Laguerre-Gauss profile

The LaguerreGaussLaser type contains the parameters required for describing a Laguerre-Gauss laser pulse. The spatial profile of the electric field is given by

\[\newcommand{\ii}{\mathrm{i}} % imaginary unit \begin{aligned} E_x(r,z,φ) &= ξ_x E_g N_{pm} \left(\frac{\sqrt{2}r}{w(z)}\right)^{|m|} ₁F₁\left(-p, |m|+1; 2\left(\frac{r}{w(z)}\right)^2\right) \exp\left[\ii\left((2p+|m|)\arctg{\frac{z}{z_R}}-mφ - \phi_0\right)\right] \\ E_y(r,z,φ) &= \frac{ξ_y}{ξ_x} E_x(r,z,φ) \\ E_z(r,z,φ) &= -\frac{\ii}{k} \left\{\left[-2\frac{1+\ii\frac{z}{z_0}}{w(z)^2} + \frac{4p}{(|m|+1)w(z)^2}\ ₁F₁^{-1}\left(-p, |m|+1; 2\left(\frac{r}{w(z)}\right)^2\right)\right](x E_x + y E_y) - \frac{|m|}{x+\ii y} (E_x \mp \ii E_y)\right\} \end{aligned}\]

where

• $E_g$ is the amplitude of the electric field form the Gaussian profile, $E_x(r,z)$
• $N_{pm}$ is a normalization factor given by $\sqrt{(p+1)_{|m|}}$, with $(x)_n$ the Pochhammer symbol
• $₁F₁(α,β;z)$ is the confluent hypergeometric function of the first kind
• $p$ is the radial index of the mode, $p ∈ ℤ, p ≥ 0$
• $m$ is the azimuthal index of the mode, $m ∈ ℤ$
• $ξ_x$ and $ξ_y$ give the polarization (choosing $ξ_x=1$ and $ξ_y=0$ for example, would give a linearly polarized field along the $x$ axis while taking them $1/\sqrt{2}$ and $±\mathrm{i}/\sqrt{2}$ would give right and left-handed circular polarization)
• $w(z)$ is the beam radius at $z$
• $w_0$ is the beam waist
• $k$ is the wavenumber
• $R$ is the radius of curvature
• $\arctg{\frac{z}{z_R}}$ is the Gouy phase
• $ϕ_0$ is the initial phase

The magnetic field field is given by

\[\newcommand{\ii}{\mathrm{i}} % imaginary unit \begin{aligned} B_x(r,z) &= -\frac{1}{c}E_y(r,z)\\ B_y(r,z) &= \frac{1}{c}E_x(r,z)\\ B_z (r,z) &= -\frac{\ii}{ck} \left\{\left[-2\frac{1+\ii\frac{z}{z_0}}{w^2(z)} + \frac{4p}{(|m|+1)w(z)^2}\ ₁F₁^{-1}\left(-p, |m|+1; 2\left(\frac{r}{w(z)}\right)^2\right)\right](y E_x + x E_y) - \frac{|m|}{x+\ii y} (E_y \mp \ii E_x)\right\} \end{aligned}\]