Density Plotting and Manipulate

Question

I have here a code. I have the density plot of a function with the parameters.

May I ask about correcting some errors. (This is a Gaussian Function)

The parameters

zr = (\[Pi]*w0^2)/\[Lambda]\
[Lambda] = 543*10^-7
wz = w0 Sqrt[1 + (z/zr)^2]
Rz = z[1 + (zr/z)^2]
theta = \[Lambda]/(\[Pi]*w0)
f = 50
w0 = 6.275*10^-1
r = sqrt[x^2 + y^2]
I (r, z) = exp[-2*[r/wz]^2]


DensityPlot[I[r, z], {x, -0.5, 0.5}, {y, -0.5, 0.5}, ColorFunction -> GrayLevel, MaxRecursion -> 5]


Color[z_] := RGBColor[0, z, 0]

Manipulate[DensityPlot[I[r, z]*(Cos[l*ArcTan[y/x]])^2, {x, -0.2, 0.2}, {y, -0.2, 0.2}, ColorFunction -> Color, MaxRecursion -> 5], {z, -50, 50}, {l, 1, 3}]

Answer 1

Your question is poorly written for this site. You should try to minimize your code and identify where the first errors are. Then ask the specific errors here on this site, we can help. We don't normally fix all the code like I have done below

I recommend you read the following

1. https://www.wolfram.com/language/fast-introduction-for-math-students/en/
2. http://www2.eng.ox.ac.uk/~labejp/Seminar/Mathematica/MathematicaSeminarNotes.pdf

But anyway, in the comments, I have identified where some of your errors were.

(* define variables before you use them *) 
\[Lambda] = 543*10^-7;
f = 50;
w0 = 6.275*10^-1;
zr = (\[Pi]*w0^2)/\[Lambda];
theta = \[Lambda]/(Pi*w0);

wz[z_] := w0 *Sqrt[1 + (z/zr)^2];
(* rz is not used*) 
rz[z_] := z (1 + (zr/z)^2)

(* mathematica functions need capitals *) 
r[x_, y_] := Sqrt[x^2 + y^2];

(* dont define your functions with capitals *) 
(* don't use [] for multiplication *) 
ifunc[x_, y_, z_] := Module[{rtemp, result},
  rtemp = r[x, y];
  result = Exp[-2*(rtemp/wz[z])^2];
  Return[result]; 
  ]

z = 1
DensityPlot[ifunc[x, y, z], {x, -0.5, 0.5}, {y, -0.5, 0.5}]
DensityPlot3D[
 ifunc[x, y, z], {x, -0.5, 0.5}, {y, -0.5, 0.5}, {z, -1, 1}]