Radial quantum number of Laguerre-Gauss modes

We introduce an operator linked with the radial index in the Laguerre-Gauss modes of a two-dimensional harmonic oscillator in cylindrical coordinates. We discuss ladder operators for this variable, and confirm that they obey the commutation relations of the su(1,1) algebra. Using this fact, we examine how basic quantum optical concepts can be recast in terms of radial modes.

Figure 1

Probability distribution $W_p$ for a coherent state $|\zeta \rangle$ with $\ell =1$ as as a function of $p$ and the ring average number $\overline{p}$.

Figure 2

Intensity profiles ${\left|\Psi (r,\varphi )\right|}^2$ for a radial coherent state $|\zeta \rangle$ written as in (3.16), with $\langle p\rangle =1$ and for an eigenstate $|p,\ell \rangle$ with $p=1$ and $\ell =1$. In the inset, we show the corresponding density plots, in the same order.

Figure 3

Intensity profiles ${\left|\Psi (r,\phi )\right|}^2$ for intelligent states with $M=10$ and $\tau =1/2$ for several values of $\ell$: $\ell =1$, $\ell =10$, and $\ell =20$. The inset shows the corresponding density plots, in the same order.

Figure 4

Intensity profiles ${\left|\Psi (r,\phi )\right|}^2$ for intelligent states with $M=11$ and $\ell =3$ for several values of $\tau$: $\tau =16/5$, $\tau =13/10$, and $\tau =3/5$. The inset shows the corresponding density plots, in the same order.