# Vortex beam profile plot [duplicate]

I want to plot this type of plot for the Lagurree Gaussian beam

\begin{aligned} u(r, \phi, z)= & C_{l p}^{L G} \frac{w_0}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{|l|} \exp \left(-\frac{r^2}{w^2(z)}\right) L_p^{|l|}\left(\frac{2 r^2}{w^2(z)}\right) \times \\ & \exp \left(-i k \frac{r^2}{2 R(z)}\right) \exp (-i l \phi) \exp (i \psi(z))\end{aligned}

• In their code there is no difference between l=1 and l=2. the spiral gap is same for both Commented May 8 at 6:58
• Then you have to formulate your question properly. Add some code and describe precisely what is not working. Commented May 8 at 7:00
• ok i will work on it. Thank you Commented May 8 at 7:03

The spiral "gaps" are not the same for l=1 and l=2 they just look so because Alex Trounev used in his code BoxRatios -> {1, 1, 1}. If you use BoxRatios -> Automatic and the same plot range in both cases you will see that the spirals are different.

(code used from Alex Trounev's anser https://mathematica.stackexchange.com/a/249405/53172)

LG[r_, ϕ_, p_, l_, w_,
z_] := (Sqrt[(2 p!)/(π (p + Abs[l])!)] 1/
w E^(-r^2/w^2) ((r Sqrt[2])/w)^Abs[l] LaguerreL[p, Abs[l],
2 r^2/w^2] E^(I l ϕ + I z));

p = 0; w = 1;

Table[ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], l u}, {u, 0,
2 Pi}, {v, -Pi, Pi}, Mesh -> None,
ColorFunction ->
Function[{x, y, z},
Hue[Abs[LG[Sqrt[x^2 + y^2], ArcTan[x, y], p, l, w, z]]]],
Boxed -> False, BoxRatios -> Automatic, Axes -> False,
PlotPoints -> 50, PlotLabel -> Row[{"l = ", l}],
ColorFunctionScaling -> False, ImageSize -> Medium,
PlotRange -> {{-1, 1}, {-1, 1}, {0, 4 Pi}}], {l, 2}] // Row