Topological phase in Laguerre-Gaussian transverse mode Why is the topological phase in a Laguerre-Gaussian transverse mode is the sum of orbital angular momenta per photon, and why is it quantized? 
 A: I take this question to mean:


*

*Why does the Laguerre-Gaussian (LG) modes have an $e^{i\ell\phi}$ dependance on the azimuthal coordinate $\phi$? 

*Why is $\ell$ required to be an integer?


Question 1
The LG modes are solutions to the paraxial wave equation in cylindrical coordinates. This means that we get solutions that reflect this symmetry. In particular the solutions should only trivially change if you make the change
$$\phi\to\phi+\Delta\phi.$$
If we define a rotation operator $R_{\Delta\phi}$ such that this operator acting on any function $f(\phi)$ gives
$$R_{\Delta\phi}f(\phi)\equiv f(\phi+\Delta\phi)$$
cylindrical symmetric solutions will be the eigenfunctions of $R_{\Delta\phi}$, i.e.
$$R_{\Delta\phi}f(\phi)=\lambda f(\phi),$$
where $\lambda$ is a constant.  The solution to this equation is of the form
$$f_\ell(\phi)\sim e^{i\ell\phi},$$
i.e.
$$R_{\Delta\phi}f_\ell(\phi)=e^{i\ell\Delta\phi}e^{i\ell\phi}=\lambda_\ell e^{i\ell\phi}.$$
Therefore cylindrically symmetric solutions such as the LG modes will be of the from
$$LG(r,\phi) = f(r)e^{i\ell\phi}.$$
Question 2
The reason $\ell$ has to be an integer (i.e. quantized) is because $\phi$ is periodic. What this means is that $\phi$ and $\phi+2\pi$ are the exact same point, therefore all functions of $\phi$ must meet the requirement
$$f(\phi+2\pi)=f(\phi).$$
If our function is $e^{i\ell\phi}$, as we saw in part 1, then this means
$$e^{i\ell(\phi+2\pi)}=e^{i\ell\phi}\to e^{i\ell 2\pi} =1,$$
which is only true if $\ell$ is an integer.
