Bessel-like beam generation by superposing multiple Airy beams

Chi-Young Hwang; Kyoung-Youm Kim; Byoungho Lee

doi:10.1364/OE.19.007356

1. Introduction

Non-diffracting beams have been a subject of active research interest and considerable amount of theoretical and experimental studies addressing their unique characteristics have been reported [1–4]. Bessel beams, whose transverse amplitude profiles show the feature of the zero-order Bessel function of the first kind were first introduced and experimentally realized by Durnin et al. in 1987 [2–5]. Since then, various forms of Bessel beams have been studied and observed [6–9].

Intuitively, non-diffracting beams like Bessel, Mathieu [10, 11], and parabolic [11, 12] beams can be synthesized by superposing conical plane waves, but these ideal types of beams are physically unrealizable because they contain infinite energy. For the experimental demonstration, Bessel beams should be truncated by finite aperture functions [11, 13]. These truncated Bessel beams carry finite energy and thus, can be realized but have limited ranges of non-diffracting propagation showing self-reconstructing behavior. There are several ways to generate Bessel beams in a laboratory: typically using an axicon lens (a cone-shaped optical element) [5, 9, 14, 15], a ring aperture [5], or adopting a holographic device [16]. Recently, Bessel beams were applied to manipulating particles [17, 18], especially to the simultaneous optical trapping in multiple planes using their self-reconstructing feature [19]. Very recently, a holographically shaped Bessel beams were introduced in microscopy to investigate inhomogeneous media [20].

In 1979, Berry and Balazs predicted non-spreading wave packet solutions in the context of quantum mechanics, called Airy wave packets [21]. Since there is an equivalence between the potential-free Schrödinger equation and the paraxial wave equation, they were introduced in the area of beam optics as well. These extraordinary wave packets, so-called Airy beams, show transversely accelerating property [22], as well as diffraction-free characteristic. But these beams also need truncation because they carry infinite power. In recent years, Siviloglou and Christodoulides presented quasi-diffraction-free finite Airy beams and showed that these beams also possess unique features of the ‘ideal’ Airy beam [23, 24]. Since then, there have been various researches on the Airy-related beams [25–29]. To produce Airy beams experimentally, we should impose cubic phase patterns to Gaussian beams using optical devices such as a spatial light modulator (SLM). Although actual observations of Airy beams were made in recent years, they were already used for various purposes: the generation of curved plasma channel [30], microparticle manipulation [31, 32], guiding Airy plasmons [33] or slow non-dispersive wave packets [34], and the formation of Airy-Bessel optical bullets [35].

In this paper, we present the theoretical formulation of Bessel-like beams (including higher-order configurations) by superposing finite-energy Airy beams which are uniformly distributed around a circle. We also report ‘vortex’ Bessel-like beams by imposing a spiral geometry to the initial launch positions of the Airy beams. Finally, based on the derived expressions, the characteristics of the Bessel-like beams are explored numerically under several conditions.

2. Theoretical formulation

The complex envelope of the Airy beam is a solution of the following (2+1)D paraxial equation of diffraction:

\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}} + i 2 k \frac{\partial ϕ}{\partial z} = 0,

where k is the wave number. We begin our derivation by assuming the following form of (2+1)D Airy wave solution [36]:

ϕ (x, y, z) = \prod_{m = x, y} u_{m} (s_{m}, ξ_{m}),

where

u_{m} (s_{m}, ξ_{m}) = A i [s_{m} + p (ξ_{m})] \exp [q (ξ_{m}) s_{m} + r (ξ_{m})],

p (ξ_{m}) = - \frac{1}{4} ξ_{m}^{2} + i a_{m} ξ_{m} + d_{m},

q (ξ_{m}) = i \frac{1}{2} ξ_{m} + a_{m},

r (ξ_{m}) = - i \frac{1}{12} ξ_{m}^{3} - \frac{1}{2} a_{m} ξ_{m}^{2} + i \frac{1}{2} (d_{m} + a_{m}^{2}) ξ_{m} .

Ai indicates the Airy function, $s_{x} = x / x_{0}$ and $s_{y} = y / y_{0}$ represent dimensionless transverse coordinates, with $x_{0}$ and $y_{0}$ arbitrary scaling factors in transverse coordinates. $ξ_{m} = z / k m_{0}^{2}$ denotes a normalized propagation distance, and $k = 2 π n / λ_{0}$ is a wave number of the optical wave in the medium having a refractive index n. $a_{m}$ is a truncation constant which has a small positive value [23, 24], and $d_{m}$ is the transverse displacement parameter. In this paper, we will assume $a_{x} = a_{y} = a$ , $d_{x} = d_{y} = d$ , and $x_{0} = y_{0} = r_{0}$ , resulting in $ξ_{x} = ξ_{y} = ξ_{r} = z / k r_{0}^{2}$ .

Through the θ-degree rotation of Eq. (2) in the transverse plane, we can get the rotated Airy beam:

\begin{array}{l} ϕ (s_{x}, s_{y}, ξ_{r}) = A i [s_{x} \cos θ + s_{y} \sin θ + p (ξ_{r})] A i [- s_{x} \sin θ + s_{y} \cos θ + p (ξ_{r})] \\ \times exp [q (ξ_{r}) {s_{x} (\cos θ - \sin θ) + s_{y} (\cos θ + \sin θ)}] \exp [2 r (ξ_{r})] . \end{array}

Expressing Eq. (5) in the cylindrical coordinate, we obtain

\begin{array}{l} ϕ (ρ, φ, ξ_{r}) = A i [ρ \cos (θ - φ) + p (ξ_{r})] A i [- ρ \sin (θ - φ) + p (ξ_{r})] \\ \times exp [q (ξ_{r}) ρ {\cos (θ - φ) - \sin (θ - φ)}] \exp [2 r (ξ_{r})], \end{array}

where

ρ = {(x^{2} + y^{2})}^{1 / 2} / r_{0}

and

\tan φ = y / x

. By integrating Eq. (6) over θ in an interval of 0 to

2 π

, i.e., by superposing multiple Airy beams having different θ values, we can get the following cylindrically symmetric beam:

ψ (ρ, ξ_{r}) = \exp [2 r (ξ_{r})] \int_{0}^{2 π} A (ρ, ξ_{r}, θ) \exp [q (ξ_{r}) ρ (\cos θ - \sin θ)] d θ,

where

A (ρ, ξ_{r}, θ) = A i [ρ \cos θ + p (ξ_{r})] A i [- ρ \sin θ + p (ξ_{r})]

. Another perspective on Eq. (7) can be achieved by assuming an ideal condition (

a = 0

) and expressing

A (ρ, ξ_{r}, θ)

as Fourier series

\sum_{n = - \infty}^{n = + \infty} A_{n} (ρ, ξ_{r}) \exp (- i n θ)

. The rearranged equation is given by

ψ (ρ, ξ_{r}) = \exp [2 r (ξ_{r})] \sum_{n = - \infty}^{+ \infty} A_{n}^{'} (ρ, ξ_{r}) \int_{0}^{2 π} \exp [- i {n θ + (2^{- 1 / 2} ρ ξ_{r}) \sin θ}] d θ .

Equation (8) suggests that the synthesized beam is a superposition of complex-amplitude modulated Bessel beam solutions.

Note that in a finite-energy case ( $a \neq 0$ ), each finite-energy Airy beam contributing to Eqs. (7) and (8) behaves like a Gaussian beam after it propagates beyond the accelerating region. Therefore, we can regard the finite-energy superimposed solution as a truncated zero-order Bessel beam. From now on, we will call these superposed beams ‘Bessel-like’ beams.

We can generate higher-order Bessel-like beams by imposing an azimuthally varying phase profile $\exp (i m θ)$ to Eq. (6), where m corresponds to the order of the Bessel-like beam and the sign of m is related to the helix handedness. These higher-order beams show the features of helical wavefronts and rotational power flow patterns in the transverse plane. It is also notable that we can make ‘vortex’ Bessel-like beams by modifying the transverse displacement parameters of each single Airy beam. In other words, in this case the parameter d in Eqs. (4a) and (4c) is a function of azimuthal angle, and this modification gives phase differences between two adjacent Airy beams.

Here, the transverse displacement parameter d determines the initial shape of the synthesized beam. The ring tails are distributed in the outside of the main ring when d has a positive value, and can exist in the inside of the main ring when d becomes negative (see Fig. 1 ). The radius of the main ring is given by $2^{1 / 2} r_{0} | d |$ . More details on these beams will be discussed with the numerical results in the following section.

Fig. 1 Calculated transverse intensity profiles of the zero-order Bessel-like beam ( $d = - 10$ ) at (a) $z = 0 μ m$ (Media 1), (b) $z = 75 μ m$ ( $ξ_{r} = 6.35$ ), and (c) $z = 150 μ m$ ( $ξ_{r} = 12.7$ ). The corresponding results for the first-order beam are shown in (d)–(f). In each figure, the intensity scale is normalized with respect to its corresponding initial distribution. (g) Radial intensity profiles of the zero-order Bessel-like beam (solid red line) shown in (c) and the ideal Bessel beam (dotted blue line). The transverse wave number of the ideal Bessel beam is suitably adjusted for the purpose of comparison. From (g), it is evident that in the far-field range, the transverse intensity pattern of the Bessel-like beam proposed here becomes similar to that of the Bessel beam.

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3. Numerical results

In order to study the characteristics of Bessel-like beams analyzed in the previous section, let us first calculate their transverse intensity profiles at several propagation distances. Figure 1 depicts the intensity plots of the zero-order (m=0) and first-order (m=1) Bessel-like beams. In these calculations, we have synthesized 120 finite-energy Airy beams using the following parameters: $λ_{0} = 532 n m$ , $n = 1$ , $r_{0} = 1 μ m$ , $a = 0.1$ , and $d = - 10$ . Note that the number of Airy beams is large enough so that Eq. (7) can be guaranteed. Therefore, we can regard the following figures as the plots of continuously superposed beams.

Although we have not presented here, when d is positive, the ring tails are distributed in the outside of the main ring, and each Airy beam seems to converge toward the propagation axis as it propagates. This feature produces a kind of auto-focusing effect [37], i.e., after a certain propagation distance, the field distributions are focused on a specific point and generate intensity peaks. We expect this focusing characteristic might be adopted for several applications.

To understand further the features of these beams, we have numerically calculated the intensity distributions in the y-z plane (x=0), as shown in Fig. 2 . Within a short propagation range ( $z < 50 μ m$ ), we can observe the curved main lobe patterns of the Airy beam. However, after a certain distance the beam intensity pattern gradually becomes that of the Bessel beam.

Fig. 2 Calculated normalized intensity profiles of (a) zero-order and (b) first-order Bessel-like beams in the y-z plane (x=0).

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As can be seen in Figs. 2(a) and 2(b), the accelerating property is maintained within the range of to ~ $50 μ m$ . In this region, these kinds of beams might be used to manipulate microparticles, especially for particle clearing (with $d < 0$ ) or gathering (having $d > 0$ ) in a symmetrical manner [38].

In Fig. 3 , we showed the detailed axial and radial intensity distributions of the zero-order and first-order Bessel-like beams under the same conditions described above. We can see that the Bessel-like beam pattern remains almost invariant up to $z = 150 μ m$ although the peak intensity is somewhat reduced.

Fig. 3 (a) Axial intensity variation of the zero-order Bessel-like beam. Radial intensity profiles of zero-order beam at (b) $z = 75 μ m$ ( $ξ_{r} = 6.35$ ) and (c) $z = 150 μ m$ ( $ξ_{r} = 12.7$ ). The corresponding results of the first-order beam are shown in (d) and (e).

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Next, we considered the propagation dynamics of Bessel-like beams having a vortex power flow. As aforementioned, they can be implemented through modifying the initial spatial distributions of multiple Airy beams in the transverse plane (z=0). Figure 4 shows the calculated transverse intensity profiles of the vortex Bessel-like beams, changing the propagation distances. The respective values of d of individual Airy beams were determined in a way that the radial distance between the center of the synthesized Bessel-like beam and that of each Airy beam increases linearly with θ. In this paper, we have arbitrarily chosen the discontinuous gap length as $6 λ_{0}$ . Note that there exist various alternative initial distributions, and the distribution shown in Figs. 4(a) and 4(d) is just one of them. It can be seen from the figures that as the beam propagates its intensity pattern shows a spiral structure in addition to its own Bessel-like pattern.

Fig. 4 Calculated transverse intensity profiles of the zero-order vortex Bessel-like beam at (a) z = 0 μm (Media 2), (b) $z = 75 μ m$ ( $ξ_{r} = 6.35$ ), and (c) $z = 150 μ m$ ( $ξ_{r} = 12.7$ ). The corresponding results of the first-order vortex Bessel-like beam are shown in (d)–(f). In each figure, the intensity scale is normalized with respect to its corresponding initial distribution.

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Lastly, we have analyzed the transverse power flow of the vortex Bessel-like beams, by numerically computing Poynting vectors $\vec{S}$ , which is given by [39]

\vec{S} = {\vec{S}}_{z} + {\vec{S}}_{⊥} = \frac{1}{2 η_{0}} {| ϕ |}^{2} \hat{z} + \frac{i}{4 η_{0} k} (ϕ \nabla_{⊥} ϕ^{*} - ϕ^{*} \nabla_{⊥} ϕ),

where

η_{0} = {(μ_{0} / ε_{0})}^{1 / 2}

is the free space impedance, and

{\vec{S}}_{z}

and

{\vec{S}}_{⊥}

indicate the longitudinal and transverse components of the Poynting vector, respectively. Numerical results are shown in Fig. 5 , with the intensity plots in the backplane of each frame. It is notable that we can change the rotational direction of the power flow by switching the handedness of geometrical distributions of constituting Airy beams in the initial plane.

Fig. 5 Calculated transverse power flow ${\vec{S}}_{⊥}$ of (a) the zero-order and (b) the first-order beams at $z = 150 μ m$ ( $ξ_{r} = 12.7$ ). Background intensity plots are magnified versions of Figs. 4(c) and 4(f).

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These Bessel-like beams can be implemented practically using the same method that is adopted for the Airy beam generation such as cubic phase modulation using an SLM. However, in this case, since we have to generate multiple Airy beams simultaneously, we need an appropriate-sized SLM.

4. Conclusion

We have presented a technique to synthesize Bessel-like non-diffracting beams (including higher-order forms) using multiple Airy beams, and demonstrated their propagation dynamics numerically. In addition, we also proposed ‘vortex’ Bessel-like beams which can be generated via a specific distribution of the individual Airy beams. By the calculation of the power flows of these vortex beams, we found that it is possible to obtain rotational power flow in the high-intensity regimes of these beams while maintaining their primary Bessel-like beam configurations. We believe these beams can be used in various applications, like optical focusing or clearing/gathering of microparticles.

Acknowledgment

The authors acknowledge the support of the National Research Foundation and the Ministry of Education, Science and Technology of Korea through the Creative Research Initiative Program (Active Plasmonics Application Systems).

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Bessel-like beam generation by superposing multiple Airy beams

Abstract

1. Introduction

2. Theoretical formulation

3. Numerical results

4. Conclusion

Acknowledgment

References and Links

Supplementary Material (2)

Cited By

Figures (5)

Equations (11)

Optics Express