Subluminal group velocity and dispersion of Laguerre Gauss beams in free space

That the speed of light in free space c is constant has been a pillar of modern physics since the derivation of Maxwell and in Einstein’s postulate in special relativity. This has been a basic assumption in light’s various applications. However, a physical beam of light has a finite extent such that even in free space it is by nature dispersive. The field confinement changes its wavevector, hence, altering the light’s group velocity vg. Here, we report the subluminal vg and consequently the dispersion in free space of Laguerre-Gauss (LG) beam, a beam known to carry orbital angular momentum. The vg of LG beam, calculated in the paraxial regime, is observed to be inversely proportional to the beam’s divergence θ0, the orbital order ℓ and the radial order p. LG beams of higher orders travel relatively slower than that of lower orders. As a consequence, LG beams of different orders separate in the temporal domain along propagation. This is an added effect to the dispersion due to field confinement. Our results are useful for treating information embedded in LG beams from astronomical sources and/or data transmission in free space.

where  L x ( ) p is the generalized Laguerre polynomial with  as integers and p ≥ 0, is the radius of curvature, and = Z kw R 1 2 0 2 is the Rayleigh length 3 . This beam spreads along propagation as illustrated in Fig. 1a. In the far-field, the beam divergence of an = =  LG p 0, 0 is represented by the opening angle θ 0 , which can be expressed in terms of minimum beam waist w 0 and magnitude of the wavevector k 0 as, In the LG expression in (1), the phase factor of the form ϕ  means  is the number of 2π windings around the azimuthal angle ϕ. First asserted by Allen et al., these beams ≠  ( 0) have Poynting vectors that spiral along the direction of propagation 4 . The helical wavefront for a beam with =  1 is illustrated in Fig. 1b. Negative s will yield the same wavefronts but of opposite helicities. The realization that LG beam carries OAM has led to a myriad of applications from optical tweezing and micromanipulation [5][6][7] , to free space information 8 , to tranverse Doppler effect 9 , and in astrophysics 10 .
Although the radial order p of LG mode is very rarely discussed, it is mostly directly used in applications. One fundamental role of p is its enhancement of the angular beam shifts in reflection of higher order LG beams 11 . Moreover, higher orders of LG modes are also found to reduce the Brownian thermal noise in laser interferometry that could be useful in future gravitational wave detectors 12 . In optical trapping, if an optical vortex due to  confines atom for precision measurements, the multi-ring dislocations due to p can be used as toroidal trap in observing persistent flow of Bose-Einstein condensates 13,14 .
In the paraxial regime, LG beams form a complete basis set such that it can be used as a tool in quantum information processes [15][16][17] . Both  and p are realized as additional degrees of freedom in encrypting information in photons 18,19 . Hence, both the orbital and radial order can be used in encoding information aside from the polarization of the light. As an application, free-space multiplexing is possible as photons are treated with higher quantum dimensional states. Consequently, higher information density can be achieved even using the same number of photons.
In this manuscript we ask: What is the effect of the orbital order  and the radial order p of LG beam on its group velocity? The consequences are extensive. The most important of which is the different time of arrival of information even in free space propagation. This is similar to the modal dispersion in fiber, a serious limitation in optical fiber communication 20 . The promised massive information when using LG beams will have an issue. Information embedded in these beams will not arrive at the same time and some corrections are then necessary.
In this paper, we report our calculation on the dispersion and reduction of v g ′s in LG beams. The analytical expression is exact and our expression reduces to the result of Giovannini et al. for Gaussian beam when =  0 and p = 0.

Results and Discussions
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, +  p 2 . When the order is zero, the beam reduces to a Gaussian mode The v g for =  0, p = 0 is consistent with the reported delay in Gaussian beams 1 . 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 ∈ −  [ 5,5] 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 =  0 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.
The v g /c becomes lower as one goes farther from =  0 and p = 0, seen by the change in the color in Fig. 2. Different orders +  p (2 ) 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 +  p 2 . 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 detection [21][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 OAM 27 . 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 α  to be dependent on  whose relation is given by, α = +   k w 1 2 2 0 0 . 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 ≠  ( 0) and p = 0, we can think that the added path length due to beam divergence increases by a factor of +  1. 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 <  0 and >  0 with respect to =  0 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 p k w ( ) 0 0 2 factor in the denominator of Equation (4). 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  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 source 28 . 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 profile 29 . 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 apertures 30,31 .
In conclusion, we have derived the group velocity v g of LG beam that is inversely proportional to the orbital order , the radial order p and the far-field beam divergence θ 0 . This result shows that LG beams are both subluminal and dispersive even in free space. Discrete ′ v s g are obtained for an arbitrary θ 0 . The dispersion of LG beams has degenerate modes for certain discrete v g ; The number of degenerate modes is just one plus the LG beam's order +  p (2 ). We also highlight that light travels in the direction of the Poynting vector, therefore both radial and angular components will contribute to the added path length. This report would have far-reaching consequences on the OAM beam's applications.

Methods
The transverse wavevector of a light beam alters both the phase velocity v p and group velocity v g . We are only concern with v g calculation since this parameter corresponds to the actual speed of light as it travels through space, whereas v p indicates the field signal variation 32 .
For a given path length Δ z between two different points such as z 1 and z 2 , a structured light travels at a time Δ t that includes an added path length δz due to the transverse components of the wavevector. They are related by Δ t = (Δ z + δz)/c. In the ray-optic model, the v g can be obtained by calculating δz and is mathematically formulated as follows,