J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 173
Fig. 1. Density plots of the spectral density of some modes, | U n( r, z)| 2, across the plane( n, z) for n = 0,1,2,3,4,5.
Using the equality J n ðtÞ ¼ 1 Z 2p e inw�it sin w dw; ð18Þ
2p 0
after some manipulations the propagated modes take the following form:
U n ðr; zÞ ¼ 2N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F pðn þ 1Þe ikz ð�iÞ nþ1 e inu e ipN F q 2 r 0
Z 1
0
J n ð2pN F quÞe ipN F u 2 u nþ1 du;
where the Fresnel number has been introduced as N F ¼ kr2 0
2pz;
together with the normalized transverse coordinate q ¼ r r 0
¼ðn; gÞ: ð19Þ ð20Þ
ð21Þ
It is seen that the vortex structure of the phase is preserved during propagation and that, once the order n has been set, the propagated field depends only on the Fresnel number, as happens for the diffraction of a plane wave by a circular hole( corresponding to n = 0)[ 61 ]. Therefore, the Fresnel number will be taken as the natural variable to represent the propagation distance. As examples, density plots of the spectral density of some modes across the plane( x, z), that is, the squared modulus of the expression in equation( 19), are shown in Figure 1.
As a first remark, we note that the property for a CSD of being uni-variable across a certain plane is not preserved during propagation. In fact, when modes propagate, their analytical forms actually change, and, in general, the product of the propagated modes, U n ðr 1; zÞ U n ðr 2; zÞ, is no longer proportional to r n 1 rn 2 exp½inðu 1 � u 2 ÞŠ, asitisrequiredfor the series to be interpreted as a power expansion.
Second, as also appears from Figure 1, the on-axis irradiance is zero for all modes, except n = 0. Furthermore, the modes spread during propagation at a rate that increases with n, giving less and less significant contributions in the central zone of the transverse planes. In particular, we expect that, when superpositions of modes are considered, only a finite number of modes need to be taken into account, which depends on the extent of the region where the field has to be evaluated. For example, modes with order 5 or higher can be safely neglected if the field has to be evaluated at N F = 0.5 within a circular region of radius 2r 0( see Fig. 1).
3.2 Coherent modes in the far field
Unlike mode propagation in the Fresnel regime, far-field conditions lead to simple analytical forms. When the Fraunhofer conditions are met, the expression of a typical field