The v g can be derived by considering geometry in the ray-optic model. The path of light follows the direction of Poynting vector which points toward the direction of the wavevector. A field confinement produces spatially structured light, which alters the wavevector to include non-axial components. The transverse components cause the delay in the v g of light. Confined light therefore, would have its v g that is not equal to c.

Suppose light travels along z in standard cylindrical coordinates (r, φ, z). A plane wave has a wavevector component that is purely along z thus, this light is expected to travel at c. For Gaussian and Bessel beams, the wavevectors comprise of both longitudinal z and radial r components. The radial component will cause an added path length in the propagation of these beams. It will generate a time delay in the speed of light. For beams with OAM, the wavevectors constitute the whole basis components. The delay then for an OAM-carrying beam is due to the added path length that originated from both radial and azimuthal wavevector components.

The v g calculation in the paraxial regime of LG beam is detailed in the Methods section. The v g is found to be inversely proportional to the orbital order , the radial order p, and the beam’s divergence θ 0 , as

This expression shows that the delay of LG beam is related to its order, . When the order is zero, the beam reduces to a Gaussian mode . The v g for , p = 0 is consistent with the reported delay in Gaussian beams1. The subluminal v g of Gaussian modes varies for different w 0 values, and that v g is even further reduced for relatively smaller w 0 . This holds true since, for a certain λ 0 , relatively lower w 0 yields larger far-field beam divergence. As the beam propagates for such case, the field confinement in the transverse structure is amplified.

For a fixed θ 0 , the expression results with discrete v g values, since and p take the values of integers and natural numbers, respectively. This fact is helpful for precise detection in communications using LG beams, as one has prior knowledge of the beams’ arrival based on discrete v g ’s.

As a representation of Equation (3), a colormap of v g /c values for and p ∈ [0, 10], is shown in Fig. 2. We generated this plot with a beam of a central wavelength λ 0 = 632.8 nm and a minimum beam waist w 0 = 2.0 μm. All values fall below unity implying subluminal speed of LG beams for any and p values. The case and p = 0, located at the center of lowest row, corresponds to v g /c of a Gaussian beam. This beam obtained the largest v g /c value or the least reduced v g . This is expected since a Gaussian beam with no radial and orbital order is the least structured beam compared to higher modes of LG beams. A Gaussian beam yields the least magnitude of transverse component in the altered wavevector, hence it intuitively results with v g closest to c.

Figure 2: Colormap of v g /c values as function of and p with central wavelength λ 0 = 632.8 nm and minimum beam waist w 0 = 2.0 μm. Each pixel of a specific color corresponds to v g /c value (colored scale bar). Warm colored pixels have relatively higher v g /c values compared to cool colored pixels. Full size image

The v g /c becomes lower as one goes farther from and p = 0, seen by the change in the color in Fig. 2. Different orders of LG beams disperse along propagation. The free-space dispersion based on Equation (3) can be expressed as the effective group index of refraction n g , given by, . For any w 0 values, n g is linearly related to . Thus, LG beams of different orders that are initially propagated simultaneously will have different time delays after travelling the same path distance. This makes LG beams separate in the temporal domain. This contributes to the dispersion due to field confinement. A beam with higher order will have greater added path length δz, evident when relating Equation (13) to Equation (10) (see derivation in Methods section).

The free-space dispersion of LG beams consequently demands corrections in their applications such as in data transmission/communication, in multiplexing, in interaction with nonlinear materials and in OAM spectrum detection21,22,23,24,25,26. The dispersion can also be substantial in quantum information processes for encryption and decryption of higher quantum dimensional states, such as and p values, in photons.

Setting p = 0 in Equation (3), the role of different values of OAM alone can be seen. Padgett et al. demonstrate that for a given beam size, the far-field opening angle increases with increasing OAM27. Larger apertures are required when receiving beams with relatively higher OAM. The -dependence of v g for LG beams that we report may be incorporated to such receiving optical system. A time-controllable receiving aperture size can be programmed according to computed delays prior to the arrival of beams. As opposed to the beam divergence relation presented in Equation (2) due to skewness of Poynting vector with respect to optical axis, they also considered the contribution of normal diffractive spreading by the standard deviation of the spatial distribution. They derived the far-field beam divergence to be dependent on whose relation is given by, . Reformulating Equation (11), the v g expression for OAM-carrying beams (p = 0) according to this beam divergence definition, we get a more compact form:

For light with OAM and p = 0, we can think that the added path length due to beam divergence increases by a factor of . This factor is consistent with the conservation of total linear momentum in the system. In the work of Giovannini et al.1, the added path length comes from the radial component of the Poynting vector with respect to the optical axis. In Equation (13) (see Methods section), we show that even a Poynting vector with angular component due to with respect to the optical axis can also contribute to the path.

Figure 3a shows the plots of v g /c versus for different p values. The symmetry of trends between and with respect to shows that the dispersion of OAM-carrying beams yield the same value of v g regardless of the helicity or polarity of . In Fig. 2, the color distributions between left and right regions mirror each other with respect to the central column, owing to the factor in Equation (3). The plot is shifted downwards for relatively higher radial order (p > 0). The v g is reduced by an added factor in the denominator of Equation (4).

Figure 3 Plots of (a) v g /c versus for different p values and (b) v g /c versus p for different values. Full size image

Similarly, v g /c is plotted against p for different values in Fig. 3b. The drop in v g /c values in these plots is steeper compared to plots of v g versus . This is due to the 2 factor in p in Equation (3). Beams of different radial orders disperse faster than beams of different OAM. The plot of v g /c versus p shifts downward as the beam is endowed with higher orbital order.

Different modes can have the same v g as seen in Fig. 3. These modes have the same beam order but of different combinations of mode indices. We call these modes with the same v g as degenerate modes. There will be more degenerate modes for lower v g . This can be seen if we include more plots for higher values of p (>3) in Fig. 3a. The same can be observed in Fig. 3b by including plots with higher , except that twice the modes must be accounted for to consider the opposite helicities. Relatively higher beam order yields more degenerate modes.

The number of degenerate modes, denoted by , in the dispersion of LG beam with order is given by,

Only the Gaussian beam is non-degenerate, which uniquely is the fastest relative to other LG modes. The number of degenerate modes is just one plus the order of the beam. Some combinations of mode indices that yield the same v g are presented in Table 1. In detection, the order of LG beam can be determined by performing cross correlation function even with intensity that resulted from partially coherent source28. There are several ways to discriminate the explicit combination of mode indices in degeneracy of the beam order. One example is to first quantify p by employing double correlation function on the captured intensity profile29. Then, the magnitude and polarity of can be characterized by measuring OAM based on Fraunhofer diffraction pattern that is formed by passing light through shaped apertures30,31.

Table 1 Some combinations of mode indices yielding degenerate v g values of LG beam in order. Full size table