The problem is that the integral b does not converge for
$l \ge 2$. Or converges very slowly, so you need to apply special calculation methods.
\[Gamma] = 1; \[Theta] = \[Pi]/2; \[Phi] = 0; l = 2; r0 = 6; w0 = 5;
Subscript[R, l] =
2/w0 (1/Sqrt[Factorial[Abs[l]]])^2*
Power[Sqrt[2] r/w0, Abs[l]]*(2*r^2/w0)*Exp[-r^2/w0^2] // Simplify;
Subscript[f, l] = Subscript[R, l]^2;
a = (1/2*Pi)*
NIntegrate[
r*Subscript[f, l]*
Exp[-344/100*Power[r/r0, 5/3]*2*r*
Abs[Sin[\[Upsilon]/2]]], {\[Upsilon], 0, 2 Pi}, {r, 0,
Infinity}]
15.6693
b = (1/2*Pi)*
NIntegrate[
r*Subscript[f, l]*
Exp[-344/100*Power[r/r0, 5/3]*2*r*Abs[Sin[\[Upsilon]/2]]]*
Exp[-2*I*l*\[Upsilon]], {\[Upsilon], 0, 2 Pi}, {r, 0, Infinity}]
NIntegrate::errprec: Catastrophic loss of precision in the global error estimate due to insufficient WorkingPrecision or divergent integral.