1. Introduction

Recently there has been a lot of attention paid to various interesting, non-trivial solutions to the paraxial wave equation. Some of these beams exhibit interesting features such as spin or orbital angular momentum[1], diffraction-free propagation [2], self reconstruction [3], or acceleration. Depending on the coordinate system there are Hermite, Laguerre, Bessel, and Ince-Gaussian solutions to name just a few. Most recently there has been increasing attention paid to the Airy solution of the paraxial wave equation first described by Berry and Balazs in 1979 [4]. The demonstration of this solution was first given in the context of the Schrödinger equation describing a free particle. The 1-D and 2-D Airy solution was most recently shown experimentally by Siviloglou et. al. in optical beams of light [5, 15]. The Airy beam is said to be free of diffractive spreading and freely accelerating in the absence of any external potential [8].

In this paper the Poynting vector and angular momentum of the Airy beam as it propagates through space is investigated. First, classical electrodynamics will be used to numerically calculate the Poynting vector of the airy wave packet. Next, two methods are employed to analyze the angular momentum of the Airy beam. The first is calculating j⃗=⃗r×p⃗ while the second is calculating the angular momentum spectrum of this beam of light [9]. These methods provide complimentary information and insight as to how this beam travels through space.

2. Airy solution to the paraxial wave equation

The (2+1)D paraxial wave equation describes the wave propagation of the electric field ϕ as,

where s_x=x/x ₀ and s_y=y/y ₀ are normalized transverse coordinates, ξ=z/k(x ₀+y ₀)² is a normalized propagation distance, x ₀ and y ₀ are normalization constants, k=2πn/λ₀ and ∇² _τ is the second partial derivative in the transverse direction.

The non-dispersive solution to this (2+1)D equation is [7],

where Ai(x) is the Airy function and is shown in the inset at the center of Fig. 1. This type of beam can be produced experimentally with a specially designed diffraction grating [5].

In theory a non-dispersive beam of this sort would have infinite power. However in practice, this can not be the case because a beam can not propagate infinite power. As discussed in Refs. [5, 6], to experimentally realize the Airy beam an initial condition must be employed to act as an exponential aperture function. Taking this initial condition as, u(s_x,s_y,0)=Ai(s_x)Ai(s_y)exp[a(s_x+s_y)], where a<<1 and is a positive parameter that limits the infinite energy in the Airy tail, the electric field amplitude becomes

Figure 1(a-d) shows the intensity in the transverse plane at various ξ-positions when a=0.15 and x_o=y_o=0.1. All of the intensity peaks are changing position as ξ increases. The main intensity peak in the s_x-s_y plane of the Airy beam travels at 45°, along the line s_x=s_y, following the trajectories,

 

Fig. 1. The intensity of the finite-energy Airy wave in the s_x-s_y plane, given by Eq. (3) with a=0.15 at ξ=0.025 (a), ξ=0.05 (b), ξ=0.075 (c), ξ=0.1 (d). The inset shows the behavior of Ai(x).

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3. The Poynting vector of Airy beams

The rate of electromagnetic energy flow per unit area, or the Poynting vector, is a commonly known quantity in electrodynamics [10, 11]. This vector is routinely examined for plane waves but has received considerable attention in the literature with regard to Laguerre-Gaussian beams of light that have helical wavefronts [1, 12, 13].

The Poynting vector is defined as [11]:

where c is the speed of light. Given a vector potential A⃗=$\widehat{\epsilon }$ u(s_x,s_y,ξ)exp[ik ² ξ(x ₀+y ₀)], where $\widehat{\epsilon }$ is an arbitrary polarization and u(s_x,s_y,ξ) is the Airy field amplitude given by Eq. (3), we can use the E⃗ and B⃗-fields in the Lorenz gauge, as given by Ref. [1], to calculate the time-averaged Poynting vector, <S⃗>. Assuming an x̂-polarized field, <S⃗> becomes [1]

The $\widehat{\xi }$ term in the above equation is the energy flow in the $\widehat{\xi }$-direction which is just proportional to the linear momentum density in that direction. This is typically the main contributing component of S⃗ in Gaussian optics. The first term is what we are really interested in here as it contributes a non-zero ŝ_x- and ŝ_y-component and an additional $\widehat{\xi }$-term to the Poynting vector.

Figure 2 (a-d) shows the numerically computed ŝ_x- and ŝ_y-components of the Poynting vector for a=0.15 at ξ=0.025, 0.050, 0.075, and 0.1, respectively. The direction and magnitude of the arrows (shown in red) correspond to the direction and magnitude of the energy flow in the transverse plane. The intensity of the Airy field is shown in the background of each frame to show the direction of the energy flow in relation to the peaks of the Airy beam. The flow of energy of the main peak at ξ>0 is consistently pointed at 45° relative to the s_x-s_y plane at all ξ-locations. In contrast, the direction of the energy flow for the Airy tails, or the peaks oriented along the horizontal or vertical axis approaches a direction perpendicular to that axis. The direction of the net energy flow is measured, however, to be constant and pointed in the direction that the main peak moves, i.e 45° or along the line s_x=s_y. This is in contrast to the locally varying direction of energy flow.

 

Fig. 2. The numerically calculated Poynting vector in the s_x-s_y plane, of the finite-energy Airy wave given by Eq. (6) with a=0.15 at ξ=0 (a), ξ=0.025 (b), ξ=0.050 (c), ξ=0.075 (d). The intensity of the Airy field is shown in the background of each frame.

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It is interesting to note that in (a) of Fig. 2, the Poynting vector is initially pointing in the negative s_x or negative s_y-direction on each Airy tail; in (b) of the figure the direction starts to turn partially towards the direction of the energy flow of the main Airy peak (45°); and in (c) and (d) as the beam propagates further, the direction swings around even more towards 45°. The change in the Poynting vector can be used to explain why in Ref. [15] the optical Airy beam is said to be able to “ascend until it stalls due to downward acceleration” since S⃗∝P⃗ which points more along the tangential directions than the ξ-direction the further the beam propagates.

If the aperture function in Eq. (3) is ignored, i.e. setting a=0, then the Poynting vector would point along the line s_x=s_y everywhere. This leads to infinite energy propagation as discussed in section 1 above and in Ref. [5]. The choice of aperture is partially responsible for the spatially varying Poynting vector yet does not seem to affect the calculations in the next section.

4. Angular momentum of the Airy beam

It is well known that as P⃗∝E⃗×B⃗ from which follows that angular momentum density about the $\widehat{\xi }$-axis is

The numerically computed <S⃗> taken from Eq. (6) and shown in Fig. 2 is used to calculate the angular momentum in the $\widehat{\xi }$-direction. Figure 3 (a-d) shows the $\widehat{\xi }$-component of the angular momentum density with a=0.15 at ξ=0.025, 0.050, 0.075, and 0.100, respectively. At ξ=0 the computed angular momentum is zero so it is not shown. In Fig. 3 reds are positive values (clockwise), blues are negative values (counter-clockwise), and green is zero. The non-discrete nature of these values will be discussed in the next section.

 

Fig. 3. The numerically calculated angular momentum in the $\widehat{\xi }$-direction of the finite-energy Airy wave given by Eq. (7) with a=0.15 at ξ=0.025 (a), ξ=0.050 (b), ξ=0.075 (c), ξ=0.100 (d). Reds are positive values, blues are negative values, and green is zero.

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As the beam propagates, the net angular momentum about the ξ-axis is always zero. The spatial distribution of the angular momentum is changing however, and locally has non-zero values of angular momentum. Not only is the angular momentum changing in the Airy tails, but there are also changes to the angular momentum in the main Airy peak. This change of angular momentum is a torque that corresponds to the force present due to the changing linear momentum. Note that changing the axis about which the angular momentum is calculated would merely alter the angular momentum by a constant.

5. Angular momentum spectrum of Airy beams

Recently a new type of imaging was proposed that is based on the phase and spatial profile of the wavefront and is coined spiral imaging[9]. This spiral imaging is similar to what Ref. [14] refers to as the orbital angular momentum (OAM) spectrum of a beam of light that they show experimentally. One can use this technique to get a more complete picture of what the zero net angular momentum density with local non-discrete, non-zero values means.

Any optical beam can be decomposed into a superposition of angular harmonics in cylindrical coordinates written as

where a_m(r,z)=1/(2π)^1/2∫^2π ₀ u(r,ϕ,z)exp(-imϕ)dϕ and the energy of each mode, m, is described by C_m=∫^∞ ₀|a_m(r,ϕ,z)|² rdr. The power, or weight, of each angular momentum state for the arbitrary field u is given by [9]

Figure 4(a-f) shows the angular momentum spectrum of the Airy beam where the field, u, is taken from Eq. (3), i.e. the weight of each spiral mode when the field is decomposed in these spiral harmonics, for ξ ranging from 0 to 0.125. Note that the sum of all of the modes at each ξ location is one and the net angular momentum is zero.

 

Fig. 4. The angular momentum spectrum of the finite-energy Airy wave given by eqs. (8, 9) with a=0.15 at ξ=0 (a), ξ=0.025 (b), ξ=0.05 (c), ξ=0.075 (d), ξ=0.100 (e), and ξ=0.125 (f).

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The angular momentum, J_ξ, is shown to have non-discrete, non-integer values (positive and negative). Figure 4 shows that this Airy field, locally, has an integer sum of discrete values of orbital angular momentum while the total angular momentum is in fact zero.

6. Conclusions

In this paper we analyze the spatial evolution of the Airy solution to the paraxial wave equation and show that while momentum is changing, energy and momentum are conserved. Looking at the change of the magnitude and direction of the airy beam can give further insight into the dynamics of this class of solution to the paraxial wave equation. This technique can be further applied to Airy beams with different “launching angles” as described by Ref. [7] which discussed the ballistic dynamics of Airy beams. When the “launching angle” of the Airy beam is varied the quantity of interest would be the change in the direction of the energy flow.

We point out that the linear and angular momenta are changing as the Airy beam propagates in the ξ-direction which should have implications when analyzing the velocity of this field. Some form of the velocity, be it phase, energy, or signal velocity, should be changing as the beam propagates and should be investigated in future studies. These beams have promise for applications in optical trapping, imaging, and spectroscopy where a sample might interact with a changing momentum and spatially varying angular momentum.