Abstract
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 v_{g}. Here, we report the subluminal v_{g} and consequently the dispersion in free space of LaguerreGauss (LG) beam, a beam known to carry orbital angular momentum. The v_{g} of LG beam, calculated in the paraxial regime, is observed to be inversely proportional to the beam’s divergence θ_{0}, 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.
Introduction
Recently, Giovannini et al. showed thru experiments, backed by calculations, that spatially structured light indeed travels slower than c within a certain path distance^{1}. That is, there is a decrease of group velocity v_{g} for structured light. Although the phenomenon can be explained classically, they used a HongOuMandel 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 space^{2}. Slowing light due to its structure is different from slowing light with materials.
LaguerreGauss (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 length^{3}. This beam spreads along propagation as illustrated in Fig. 1a. In the farfield, the beam divergence of an 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 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 propagation^{4}. 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 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 multiring dislocations due to p can be used as toroidal trap in observing persistent flow of BoseEinstein 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, freespace 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 and p = 0.
Results and Discussions
The v_{g} can be derived by considering geometry in the rayoptic 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 nonaxial 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 OAMcarrying 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 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 farfield 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.
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 freespace 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 freespace 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 farfield 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 timecontrollable 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 farfield beam divergence to be dependent on whose relation is given by, . Reformulating Equation (11), the v_{g} expression for OAMcarrying 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 OAMcarrying 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).
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 nondegenerate, 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 farfield beam divergence θ_{0}. This result shows that LG beams are both subluminal and dispersive even in free space. Discrete 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 . 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 farreaching 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 Δz between two different points such as z_{1} and z_{2}, 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 rayoptic model, the v_{g} 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 v_{g} 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 zindependent v_{g} for any arbitrary field. This then translates spatial dependence of v_{g} 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 z_{1} to z_{2}:
where is the operator representing the transverse wave vector. By taking an inner product of ψ(z_{2}, 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/k_{0}w_{0} is replaced by the opening angle of the beam θ_{0}/4 for a more intuitive picture. When and p = 0,
Equation (15) is consistent with the calculation for Gaussian beam^{1}.
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).
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Acknowledgements
This work is supported by the University of the Philippines Office of the VicePresident for Academic Affairs thru its BalikPhD program (OVPAABPhD 201506).
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N.H. and N.D.B. equally contributed to the research. N.H. provided the initial problem. N.D.B. made the calculations. Both contributed to the analysis of the results.
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Bareza, N., Hermosa, N. Subluminal group velocity and dispersion of Laguerre Gauss beams in free space. Sci Rep 6, 26842 (2016). https://doi.org/10.1038/srep26842
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