Introduction

Recently, Giovannini et al. showed thru experiments, backed by calculations, that spatially structured light indeed travels slower than c within a certain path distance1. That is, there is a decrease of group velocity vg for structured light. Although the phenomenon can be explained classically, they used a Hong-Ou-Mandel interferometer to measure the lag of a laterally structured photon compared to a photon with little lateral structure. In their experiment, the slowing of light is due to dispersion in free space. They performed their experiment with a Bessel beam and a Gaussian beam. Alfano and Nolan remarked that by considering dispersion relation, Bessel beam can be very slow near a critical frequency which can be used as optical buffer in free space2. Slowing light due to its structure is different from slowing light with materials.

Laguerre-Gauss (LG) beam is an interesting structured light since it carries orbital angular momentum (OAM). LG beam can have orders of orbital or winding order and radial order p. The scalar field of LG beam is expressed mathematically in standard cylindrical coordinates (r, φ, z) as follows,

where is the generalized Laguerre polynomial with as integers and p ≥ 0, is the beam waist, is the radius of curvature, and is the Rayleigh length3. This beam spreads along propagation as illustrated in Fig. 1a. In the far-field, the beam divergence of an is represented by the opening angle θ0, which can be expressed in terms of minimum beam waist w0 and magnitude of the wavevector k0 as,

Figure 1
figure 1

(a) LG beam spreading through propagation and (b) LG wavefront for and p = 0.

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In the LG expression in (1), the phase factor of the form means is the number of 2π windings around the azimuthal angle φ. First asserted by Allen et al., these beams have Poynting vectors that spiral along the direction of propagation4. The helical wavefront for a beam with is illustrated in Fig. 1b. Negative 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 micromanipulation5,6,7, to free space information8, to tranverse Doppler effect9, and in astrophysics10.

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 beams11. 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 detectors12. 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 condensates13,14.

In the paraxial regime, LG beams form a complete basis set such that it can be used as a tool in quantum information processes15,16,17. Both and p are realized as additional degrees of freedom in encrypting information in photons18,19. 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 communication20. 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 vg′s in LG beams. The analytical expression is exact and our expression reduces to the result of Giovannini et al. for Gaussian beam when and p = 0.

Results and Discussions

The vg 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 vg of light. Confined light therefore, would have its vg 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 vg calculation in the paraxial regime of LG beam is detailed in the Methods section. The vg is found to be inversely proportional to the orbital order , the radial order p, and the beam’s divergence θ0, 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 vg for , p = 0 is consistent with the reported delay in Gaussian beams1. The subluminal vg of Gaussian modes varies for different w0 values, and that vg is even further reduced for relatively smaller w0. This holds true since, for a certain λ0, relatively lower w0 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 θ0, the expression results with discrete vg 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 vg’s.

As a representation of Equation (3), a colormap of vg/c values for and p ∈ [0, 10], is shown in Fig. 2. We generated this plot with a beam of a central wavelength λ0 = 632.8 nm and a minimum beam waist w0 = 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 vg/c of a Gaussian beam. This beam obtained the largest vg/c value or the least reduced vg. 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 vg closest to c.

Figure 2: Colormap of vg/c values as function of and p with central wavelength λ0 = 632.8 nm and minimum beam waist w0 = 2.0 μm.
figure 2

Each pixel of a specific color corresponds to vg/c value (colored scale bar). Warm colored pixels have relatively higher vg/c values compared to cool colored pixels.

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The vg/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 ng, given by, . For any w0 values, ng 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 vg 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 vg 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 vg/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 vg 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 vg is reduced by an added factor in the denominator of Equation (4).

Figure 3
figure 3

Plots of (a) vg/c versus for different p values and (b) vg/c versus p for different values.

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Similarly, vg/c is plotted against p for different values in Fig. 3b. The drop in vg/c values in these plots is steeper compared to plots of vg 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 vg/c versus p shifts downward as the beam is endowed with higher orbital order.

Different modes can have the same vg 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 vg as degenerate modes. There will be more degenerate modes for lower vg. 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 vg 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 vg values of LG beam in order.
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In conclusion, we have derived the group velocity vg of LG beam that is inversely proportional to the orbital order , the radial order p and the far-field beam divergence θ0. This result shows that LG beams are both subluminal and dispersive even in free space. Discrete are obtained for an arbitrary θ0. The dispersion of LG beams has degenerate modes for certain discrete vg; The number of degenerate modes is just one plus the LG beam’s order . 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 vp and group velocity vg. We are only concern with vg calculation since this parameter corresponds to the actual speed of light as it travels through space, whereas vp indicates the field signal variation32.

For a given path length Δz between two different points such as z1 and z2, a structured light travels at a time Δt that includes an added path length δz due to the transverse components of the wavevector. They are related by Δt = (Δz + δz)/c. In the ray-optic model, the vg can be obtained by calculating δz and is mathematically formulated as follows,

The path of light can be represented as a diverted ray with a certain angle from the beam axis. The amount of δz is the difference between length of diverted ray within the actual path and Δz. This can be expressed as

where Φ is the phase profile of the scalar field. As an example, Φ = kz − ωt for plane wave thus δz is zero, as expected. However for LG beam, the beam waist varies significantly at distances near the Rayleigh length. This manifests variation of vg as it propagates in the near field. We consider the paraxial regime in order to simplify further δz, so that we can derive an expression of z-independent vg for any arbitrary field. This then translates spatial dependence of vg into wavevector. The derivation by Giovannini et al.1 considers Φ as complex argument of the scalar field function,

The paraxial wave equation is then written in terms of quantum mechanical operator with evolution of wavefunction from z1 to z2:

where is the operator representing the transverse wave vector. By taking an inner product of ψ(z2, k) in Equation (9) and substituting the result to Equation (7), we obtain the relation:

so that Equation (6) becomes,

where,

such that operator is the transverse Laplacian.

Now, for an LG beam, we substitute Equation (1) to Equation (12) in order to have

And the group velocity for such beam is given by,

where 1/k0w0 is replaced by the opening angle of the beam θ0/4 for a more intuitive picture. When and p = 0,

Equation (15) is consistent with the calculation for Gaussian beam1.

Additional Information

How to cite this article: Bareza, N. D. and Hermosa, N. Subluminal group velocity and dispersion of Laguerre Gauss beams in free space. Sci. Rep. 6, 26842; doi: 10.1038/srep26842 (2016).