Lost and found: the 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. 
E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Rehacek, Z.   Hradil, A. Aiello, G. Leuchs and L. L. Sanchez-Soto
quant-ph