Three-dimensional localized Airy-Laguerre-Gaussian wave packets in free space

Wei-Ping Zhong; Milivoj Belić; Yiqi Zhang

doi:10.1364/OE.23.023867

1. Introduction

Airy wave packet, one of the non-spreading linear beams, is the nontrivial one-dimensional (1D) solution of the potential-free Schrödinger equation. A remarkable characteristic of this wave packet is that it freely accelerates, even in the absence of any external potential. In fact, the center of “mass” of the beam does not accelerate, but the major lobes or caustics of the beam do. First suggested more than 30 years ago in the pioneering work by Berry and Balazs [1], the research on accelerating linear beams has been extended to various fields of optics. Airy beam has attracted intense attention due to its unique properties, such as self-acceleration [2], ballistic dynamics [3], and self-healing [4]. However, the full non-spreading Airy wave has infinite energy, which is not a realistic physical situation. It was not until 2007 that the finite-energy Airy wave packet was introduced by using an exponential aperture function [2,5]. In contradistinction to Bessel beams [6,7], the Airy beams do not rely on conical superposition of plane waves and possess properties of self-acceleration, in addition to non-spreading and self-healing.

The extension from 1D to 2D Airy beams leads to a number of systematic explorations on the generation, manipulation, and general appearance of Airy beams in linear [8–10] and nonlinear media [11,12]. At present, spatiotemporal optical wave packets or light bullets containing both dispersion and diffraction, have been explored by many researchers. Abdollahpour et al. demonstrated the realization of a triple Airy configuration, by combining two spatial Airy beams with an Airy pulse in time [13]. A particularly interesting advance in 3D was to combine 1D Airy beam with a 2D Bessel beam, to form spatiotemporal Airy-Bessel light bullets [9]. For Airy-Bessel light bullets, the Bessel function does not converge (even the zero-order Bessel function) – its energy is infinite in the transverse direction and considerable apodization is required. Owing to perceived versatility and potential applicability of such beams, we reconsider here their generation under more general and more realistic conditions. Namely, we incorporate the decay factor into the Airy component of the wave packet and combine two counter-accelerating finite energy Airy wave functions, to produce a compressed beam with controllable finite energy in time that does not accelerate. In addition, instead of Bessel functions we consider the generalized Laguerre-Gaussian polynomials with convergent features in the transverse coordinate R, to produce finite-energy Airy-Laguerre-Gaussian beams in different forms, including the fundamental disks, vortex rings, azimuthally-modulated rings, and the symmetric necklace structures.

Thus, in this paper we explore a model based on the spatiotemporal linear Schrödinger equation without external potential, to investigate the propagation of 3D Airy-Laguerre-Gaussian wave packets in free space or a linear medium. An exact solution containing a superposition of two counter-accelerating Airy functions and the generalized Laguerre-Gaussian polynomials is obtained by applying the method of separation of variables. We find that the corresponding optical intensity can be manipulated by four parameters: the decay factor, the radial mode number, the azimuthal mode number, and the modulation depth. Different from our previous studies [14,15], where the combination of two counter-accelerating Airy wave functions was used to manipulate only the temporal properties using the decay factor, with the aid of the generalized Laguerre-Gaussian polynomials in polar coordinates, the controlling freedom of the spatial properties is enhanced. To the best of our knowledge, the linear Airy-Laguerre-Gaussian wave packets have not been reported yet. Our results will hopefully enrich the understanding of linear localized structures of the potential-free Schrödinger equation in physics and may provide useful information for potential applications of such freely propagating beam systems over several Rayleigh lengths without appreciable change in their properties. These properties may turn useful in potential applications, such as optical tweezing [16], the generation of plasma channels [17], and microlithography [18].

The organization of the paper is as follows. In Sec. 2, an exact 3D Airy-Laguerre-Gaussian linear wave packet solution is reported in free space, without any external potential. Different profiles of 3D wave packets are probed at different propagation distances in Sec. 3, for which there are four relevant parameters. We also compare our exact analytical results with the numerical solutions in Sec. 3, by using the split-step beam-propagation method. The paper is concluded in Sec. 4.

2. The model and the 3D Airy-Laguerre-Gaussian solution

To introduce a new class of 3D Airy-Laguerre-Gaussian wave packet solutions, we consider a 3D dispersive and diffractive optical paraxial system. This situation can arise in normally dispersive planar waveguides, where diffraction is two-dimensional in the transverse plane. To simplify analysis, we equalize dispersion and diffraction effects and assume anomalous dispersion. Spatiotemporal evolution of the 3D wave packet is then described by the paraxial wave equation, which is equivalent to the (3 + 1)D potential-free Schrödinger equation [2,3,9]

i \frac{\partial u}{\partial ζ} + \frac{1}{2} (\frac{\partial^{2} u}{\partial R^{2}} + \frac{1}{R} \frac{\partial u}{\partial R} + \frac{1}{R^{2}} \frac{\partial^{2} u}{\partial ϕ^{2}} + \frac{\partial^{2} u}{\partial T^{2}}) = 0

where

u (ζ, R, ϕ, T)

is the complex envelope of the optical field, and

R = \sqrt{x^{2} + y^{2}} = r / r_{0}

and

T = t / t_{0}

represent the dimensionless transverse coordinate and the retarded time in the reference frame moving with the optical pulse. Here

r_{0}

and

t_{0}

are arbitrary scaling parameters. The longitudinal propagation distance

ζ = z / k r_{0}^{2}

is measured in terms of the corresponding Rayleigh length, where

k = 2 π N / λ_{0}

is the wave number with the wavelength λ₀ and the refractive index N; φ is the azimuthal angle in the transverse polar coordinates. The second, third and fourth terms in Eq. (1) describe diffraction and the last term originates from the optical dispersion.

To obtain linear wave packet solution of Eq. (1), we use the method of separation of variables. We assume a solution of the form $u (ζ, R, ϕ, T) = V (ζ, R) Φ (ϕ) P (ζ, T)$ and upon substitution obtain the following three partial differential equations (PDEs):

i \frac{\partial P}{\partial ζ} + \frac{1}{2} \frac{\partial^{2} P}{\partial T^{2}} = 0

\frac{\partial^{2} Φ}{\partial ϕ^{2}} + m^{2} Φ = 0

i \frac{\partial V}{\partial ζ} + \frac{1}{2} (\frac{\partial^{2} V}{\partial R^{2}} + \frac{1}{R} \frac{\partial V}{\partial R} - \frac{m^{2}}{R^{2}} V) = 0

where m is the azimuthal mode number (also known as the topological charge), which is an integer, as discussed previously [19–24]. We first focus on Eq. (2)A) and specifically investigate the dynamics of finite-energy Airy beams in the temporal domain T, by considering a specific input into the system (at ζ = 0) of the form P(ζ = 0,T) = Ai(σT)exp(σaT), where σ = ± 1, Ai(·) is the Airy function, and a (0<a<1) is the decay parameter [2,3].

By solving the linear parabolic PDE (2A) under such an initial condition, one still may obtain different particular solutions; for our purposes the counter-accelerating beams for σ = 1 and σ = −1 are especially useful:

P_{+} (ζ, T) = Ai (T - \frac{ζ^{2}}{4} + i a ζ) e^{a T - \frac{1}{4} a ζ^{2} + i (- \frac{1}{24} ζ^{3} + \frac{1}{2} a^{2} ζ + \frac{1}{2} T ζ)}

P_{-} (ζ, T) = Ai (- T - \frac{ζ^{2}}{4} + i a ζ) e^{- a T - \frac{1}{4} a ζ^{2} + i (- \frac{1}{24} ζ^{3} + \frac{1}{2} a^{2} ζ - \frac{1}{2} T ζ)}

Thus, we construct the following solution by superposing the two counter-accelerating Airy wave functions, i.e.,

P (ζ, T) = P_{+} (ζ, T) + i P_{-} (ζ, T)

Obviously, Eq. (3) is also a solution of Eq. (2)A). Equations (3)A) and (3B) show that the intensity profiles of P₊(ζ,T) and P_-(ζ,T) remain invariant over extended intervals. Figure 1 displays some beam profiles which involve an individual and a superposition of two counter-accelerating Airy beams. Left panels in Fig. 1 display the intensity profiles at various distances. In Fig. 1(a), for σ = 1, the beam is accelerating with propagation distance in the positive T direction, as shown by the red arrow; for σ = −1, as shown in Fig. 1(b), the beam accelerates in the negative T direction, as indicated by the blue arrow. One can note that upon propagation, the individual beams attenuate slowly and acquire long flat tails. However, if one considers their superposition, the peak intensity in the tail first increases and then attenuates fast, as seen in the left panel of Fig. 1(c). Thus, in the region of interest, the superposed beam does not accelerate and appears compressed, which is the reason for considering such a superposition in the first place. An intensity of this type effectively arises from the collision of two counter-accelerating beams, which creates a non-accelerating wave packet. After the collision, the beams accelerate separately in the opposite directions without affecting each other in this linear system. However, by the time they separate, their utility as diffraction-free accelerating beams is already diminished. For the given parameters, the intensity profile of the beam remains essentially invariant up to 63 cm, which is equivalent to 5 Rayleigh lengths. Evidently, this wave packet endures because of the dispersion-free character of the Airy wave packet. The beam will effectively lose its Airy beam properties at the propagation distances z≥100 cm (ζ≥8) and become a broad featureless long pulse with slight peaks in both tails, as exhibited in the left column of Fig. 1(c).

Fig. 1 (a) Accelerating finite-energy Airy beam with σ = 1 and I₊ = |P₊|², along the positive t direction. (b) Accelerating finite-energy Airy beam with σ = −1 and I_- = |P_-|², along the negative t direction. (c) Non-accelerating superposition of the two counter-accelerating beams. The left column: Intensity profiles at various distances. The right column: Contour-plots of the beam intensity distribution, as functions of the propagation distance. The parameters are: r₀ = 100 μm, λ₀ = 0.5 μm, and a = 0.15. Note that ζ = 5 when z = 63 cm.

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Next, we consider the solution of the azimuthal equation. Obviously, Eq. (2)B) has an analytical solution [25,26]

Φ (ϕ) = \cos (m ϕ) + i q \sin (m ϕ)

where the parameter

q \in [0, 1]

determines the modulation depth of the beam intensity [27,28].

Finally, we turn to Eq. (2)C), which is a typical paraxial wave equation describing an optical beam propagation in 2D free space. According to our previous self-similar solution results [27,28], the solution of Eq. (2)C) can be written as

V (ζ, R) = \sqrt{\frac{n!}{(n + m)!}} \frac{w_{0}}{w (ζ)} {[\frac{\sqrt{2} R}{w (ζ)}]}^{m} L_{n}^{m} [\frac{2 R^{2}}{w^{2} (ζ)}] e^{- \frac{R^{2}}{2 w^{2} (ζ)} + i [- \frac{ζ}{2 (ζ^{2} + ζ_{R}^{2})} R^{2} - (2 n - m + 1) arctan (\frac{ζ}{ζ_{R}})]}

where n is a non-negative integer which is called the radial mode number, w₀ is the beam waist of the fundamental or Gaussian mode (n = 0, m = 0),

w (ζ) = w_{0} \sqrt{1 + ζ^{2} / ζ_{R}^{2}}

is the beam width at the propagation distance

ζ

,

ζ_{R}

is the Rayleigh length, which is the unit of length in the propagation

ζ

direction, and

L_{n}^{m} [2 R^{2} / w^{2} (ζ)]

is the generalized Laguerre polynomial of the n-th order [29]. Equation (2)C) can be treated with other methods, but the same results are obtained [30,31].

The complete class of wave packet solutions of Eq. (1) can be readily constructed using Eqs. (3)-(5), as

u (ζ, T, r, ϕ) = V (ζ, R) (P_{+} (ζ, T) + i P_{-} (ζ, T)) [\sin (m ϕ) + i q \cos (m ϕ)]

where

P_{+} (ζ, T)

,

P_{-} (ζ, T)

and

V (ζ, R)

are determined by Eqs. (3)A), (3B) and (5). Clearly, the modal solution (6) to Eq. (1) satisfies |u|→0 as |r|→∞, i.e., Eq. (9) represents a localized linear beam solution. In the next section, we analyze its properties.

3. Dynamical evolution of the localized Airy-Laguerre-Gaussian wave packets

Based on the analytical solution (6) derived in the last section, we discuss the propagation properties of the linear optical wave packets at two observation points: ζ = 0 (z = 0 cm) and ζ = 3ζ_R (z = 37.8 cm). The calculation parameters are chosen as w₀ = 1 and ζ_R = 1. We exhibit the 3D contour profiles of the intensity distribution of these linear beams. The localized optical wave packets from Eq. (6) can be conveniently classified by four parameters, the radial mode number n, the azimuthal mode number m, the modulation depth q, and the decay factor a. By selecting different values of these parameters, one obtains interesting examples of controllable Airy-Laguerre-Gaussian beams.

After presenting some novel profiles of analytical linear wave packets, we verify that the solutions of Eq. (1), as given by Eq. (6), do not collapse or become unstable, on the account of Eq. (1) being a linear equation. Indeed, the comparison of analytical solutions with numerical simulations of the 3D potential-free Schrödinger equations shows that the analytical solutions are in good agreement with the numerical simulations.

3.1. Zero-vorticity Airy-Laguerre-Gaussian beams (m = 0)

When m = 0, as seen from Eqs. (5) and (6), the intensity of the beam is independent of the azimuthal mode number. It is easy to see that analytical solution (6) yields a cluster of zero-vorticity wave packets with layered and multilayered structures for different values of n, as shown in Fig. 2 at ζ = 0 (the left column) and ζ = 3ζ_R (the right column) with the decay factor a = 0.15. When n = 0, the optical wave packet forms a 3D cluster of ellipsoids that represent the fundamental Gaussian beam (here the fundamental state is not displayed) with the finite-energy Airy pulse; a similar result is shown in [27]. The first excited state occurs when n = 1; an isosurface intensity contour plot of this beam is exhibited in Fig. 2 (a). It features a train of seven ellipsoidal pulses stacked along the T-axis (the vertical direction). The largest middle ellipsoidal pulse in the plane T = 0 is encircled by one ring pulse. When n increases from 1 to 2, the second excited state becomes more complicated. The largest disk pulse is surrounded by two coaxial rings at T = 0 and the disk pluses below and above are encircled by a ring at T = −2 and T = 2 planes, respectively. The two wave packets accelerate oppositely along the T-axis, as shown by the red and blue arrows, but the resulting wave packet does not accelerate. Generally, for a positive integer n, there exists an ellipsoid girthed by n rings. The larger the value of n, the thicker the ring. The maximum optical intensity is located at the central position (x,y,T) = (0,0,0), as expected.

Fig. 2 Snapshots describing the evolution of a linear beam with zero-vorticity (m = 0) at ζ = 0 (the left column) and ζ = 3ζ_R (the right column) with n = 1 (top row), n = 2 (bottom row) and a = 0.15.

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3.2. Vortex Airy-Laguerre-Gaussian beams (q = 1, m≠0)

For a multidimensional optical wave packet, the phase singularity may form an optical vortex: the wave rotates around the vortex core in a given direction. At the center, the velocity of this rotation is infinite and the light intensity vanishes. The unique, robust nature of vortex fields is expected to lead to applications in areas that include optical data storage, distribution, and processing [32]. Let us now discuss the general case when q = 1 and m≠0. As it will be seen, this case leads the transverse field to vanish at the central position. Clearly, as visible in Eq. (5), when R = 0 the intensity of the beam becomes zero, because ${[\sqrt{2} R / w (ζ)]}^{m} = 0$ ; this 3D beam appears as a vortex.Next, we analyze the pattern distributions of 3D vortex beams in space. Some vortex profiles with a = 0.1 are displayed in Fig. 3 for different $n$ and $m$ . Figure 3(a) presents the intensity of the wave packet with n = 1 and m = 1. The major difference from Fig. 2(a) is readily apparent – the central ellipsoidal pulses change into rings. We consider more complex cases, for example, n = 3 and m = 1; Fig. 3(b) shows the distribution of the vortex wave packet. As seen, three coaxial rings have emerged at the positions T = −2, T = 0 and T = 2, and two coaxial rings at T = −4 and T = 4. The smaller the absolute value of T, the thicker the ring in the horizontal (x,y) plane. If a higher value of m is chosen, similar patterns will be displayed, but with more rings encircling the T-axis.

Fig. 3 Vortex Airy-Laguerre-Gaussian beams. Setup is the same as in Fig. 2, except for n = 3 in the bottom row, m = 1, and a = 0.1.

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3.3. Necklace Airy-Laguerre-Gaussian beams (q = 0, m≠0)

The final possible case of the cluster solutions given by Eq. (6) is obtained when q = 0 and m≠0. An isosurface contour plot of this wave packet is depicted in Fig. 4. The intensity structure involves 3D necklace beams, as shown in Fig. 4(a), which displays a structure with seven layers in the vertical direction (T-axis), and ten necklace ellipsoids in the horizontal (x,y) plane in each layer, due to the azimuthal modulation. Here, n = 0, m = 5, and a = 0.2. Figure 4(b), shows another isosurface plot of this localized necklace wave, with n = 1, m = 3, and a = 0.2.

Fig. 4 Necklace Airy-Laguerre-Gaussian beams at ζ = 0 (the left column) and ζ = 3ζ_R (the right column). The parameters are (a) n = 0, m = 5 (top row); (b) n = 1, m = 3 (bottom row), and a = 0.2. Color is used for easier identification of different pulse clusters.

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3.4. Comparison with the numerical solution of the (3 + 1)D potential-free Schrödinger equation

The validity of the analytical solution can be verified by the direct numerical simulation of Eq. (1). The aim is not to check the stability of the analytical solution – as a solution of the linear partial differential equation it surely is – but to check the development of an initial exact solution in numerical simulation. Figure 5 shows the comparison between analytical and numerical solutions of Eq. (1), with the parameters: a = 0.15, n = 3, m = 0. Numerical simulation is executed by means of the split-step beam-propagation method [33]. Figure 5(a) displays the initial beam profile of the analytical solution (6) at ζ = 0, Fig. 5(b) is the pattern of the analytical solution of Eq. (6) at ζ = 3ζ_R, while Fig. 5(c) illustrates the numerical simulation. One can see that the analytical solution is consistent with the numerical result, which is not unexpected for this linear optical system.

Fig. 5 Comparison of the analytical solution with the numerical simulation for the parameters a = 0.15, n = 3. (a) The initial beam profile, (b) the analytical solution (6) at ζ = 3ζ_R, (c) the numerical simulation at ζ = 3ζ_R.

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At the end, we present some conditions on the real physical systems utilized in experimental research that might be useful for actual demonstration of these beams. Specifically, the beam should propagate in an anomalous dispersion material, and the diffraction and dispersion coefficients should be arranged so that Eq. (1) is valid, i.e., the diffractive and dispersive effects should contribute equally to the propagation of the wave packet. This requirement is not too stringent, amounting to the request that the product of the wave number k and the dispersion coefficient β₂ = d²k/dω² should equal (t₀/r₀)², where r₀ and t₀ are the arbitrary scaling parameters introduced after Eq. (1). Airy-Laguerre-Gaussian wave packets can be studied experimentally in anomalously dispersive bulk media, such as silica glass. Silica, at the wavelength 1550nm, exhibits a dispersive coefficient of $- 2.8 \times 10^{- 2} p s^{2} / m$ . For this example, the formal equalization of dispersion and diffraction effects could sustain stable propagation of the linear wave packet over extended distances.

4. Conclusion

In summary, we have introduced Airy-Laguerre-Gaussian linear wave packets, which are described by the (3 + 1)D potential-free Schrödinger equation. These beam solutions are constructed with the aid of the superposition of the finite energy Airy wave functions and the generalized Laguerre-Gaussian polynomials. The dynamics of such wave packets was analytically studied when four different parameters are varied: the radial mode number, the azimuthal mode number, the modulation depth, and the decay factor. The validity of our analytical solution was confirmed by numerical simulations. Our results show that the Airy-Laguerre-Gaussian beams may display the fundamental disk shape, vortex rings, azimuthally-modulated rings, and the symmetric necklace structures.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61275001) and by the Natural Science Foundation of Guangdong Province, China (No. 2014A030313799). The work at the Texas A&M University at Qatar was supported by the NPRP 6-021-1-005 project with the Qatar National Research Fund (a member of the Qatar Foundation).

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Three-dimensional localized Airy-Laguerre-Gaussian wave packets in free space

Abstract

1. Introduction

2. The model and the 3D Airy-Laguerre-Gaussian solution

3. Dynamical evolution of the localized Airy-Laguerre-Gaussian wave packets

3.1. Zero-vorticity Airy-Laguerre-Gaussian beams (m = 0)

3.2. Vortex Airy-Laguerre-Gaussian beams (q = 1, m≠0)

3.3. Necklace Airy-Laguerre-Gaussian beams (q = 0, m≠0)

3.4. Comparison with the numerical solution of the (3 + 1)D potential-free Schrödinger equation

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (10)

Optics Express