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.