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

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

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

1 Answer 1


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

enter image description here