Quasi-Airy beams along tunable propagation trajectories and directions

Yixian Qian; Site Zhang

doi:10.1364/OE.24.009489

1. Introduction

Airy beam was first predicted theoretically by Berry and Balazs [1] as exact solutions of the Schrödinger equation, which was observed physically by a decaying version by Siviloglou and Christodoulides in 2007 [2]. Recently, Airy beam has attracted considerable attention owing to its intriguing properties, such as non-diffraction [3], self-acceleration [4] and self-healing [5] during propagation. These special properties have exhibited different applications in atom manipulating [6–8], optical clearing [9], and self-bending plasma channels [10,11]. Perhaps the most fantastic property is the propagation along parabolic trajectory. Such self-accelerating property facilitates to optically move a particle in air or liquid along any desired trajectory or get around any obstacle. Miguel A. Bandres demonstrated accelerating parabolic Airy beams, which are the only orthogonal and complete family of explicit solutions of the two-dimensional paraxial wave equation [12]. Lately, non-broadening optical beams propagating only along convex trajectories were reported [13,14], and they also display the same universal intensity cross section as an Airy beam, irrespective of their acceleration. Recently, Yan and Yao proposed an accelerating nondiffracting beams [15], which combine the properties of accelerating beams and diffraction-free beams. Yi Hu et al proposed an optimal control of the ballistic motion of Airy beams by deviating a Gaussian beam or phase mask from the optical axis [16]. In this way, the mainlobe intensity of Airy beam can move along a predesigned trajectory. However, it is not convenient due to adjusting the Gaussian beam and phase manually. Moreover, in the vast majority of prior research [17,18], even if the propagation trajectory can be tunable, the propagation directions always spread along the fixed $45 °$ or limitative direction on the $x - y$ plane.

In this letter, we investigate and analyze accelerating quasi-Airy beams theoretically and experimentally, which are generated by introducing two disturbance factors into cubic phase function. In addition to generating a new family of Airy beam, the benefit is that the main lobe of generated quasi-Airy beam can propagate along any predesigned trajectories and directions by adjusting readily the angle of side lobes, furthermore, the intensity distributions of quasi-Airy beam are distinctly different that of Airy beam.

The main lobe intensity of quasi-Airy beam can be “delivered” to anywhere along the known trajectory. This brings about the possibility to send an intense laser beam to a target while avoiding turbulent medium or obstacles. The advantage can also facilitate the quasi-Airy beam to evade obstacle and transport a particle to a desired location in optical tweezers. Their optical dynamics are discussed, and we demonstrate that they also have the ability of self-healing and non-diffraction similar to the Airy beam. However, the ability depends on the included angle formed by the two sidelobes. The experiment results are consistent with the numerical simulations.

2. Theory

To investigate the propagation dynamics of the quasi-Airy beam, first we employ the two-dimensional (2D) lineal potential Schrodinger equation in quantum mechanics for the electric field $ϕ (x, y, z)$ ,

i \frac{\partial ϕ}{\partial z} + \frac{1}{2 k} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) = 0.

Where

ϕ

represents the electric field envelope.

k = 2 π n / λ_{0}

denotes the wavenumber of the optical wave, and z is the propagation distance. However, the Airy beam cannot be physically realized due to its infinite power. By introducing an exponentially decaying factor

\exp (a s)

[3], here s denotes x or y, the solution of finite energy of Eq. (1) can be expressed as follows:

\begin{array}{l} ϕ (x, y, z) = A i [x - (^{z / 2) 2} + i a z] \exp [a x - (a z^{2} / 2) - i (z^{3} / 12) + i (a^{2} z / 2) + i (x z / 2)) \\ _{}^{}_{}^{}_{}^{}_{}^{} \times A i [y - (^{z / 2) 2} + i a z] \exp [a y - (a z^{2} / 2) - i (z^{3} / 12) + i (a^{2} z / 2) + i (y z / 2)) . \end{array}

Where $A i (\cdot)$ denotes Airy function. The parameter a is a small positive quantity, which can suppress the inﬁnite Airy tail and thus ensure the physical realization of Airy beams. However, what we are more interested in is the Fourier spectrum of the finite-energy Airy beam, which is expressed in $k_{x} - k_{y}$ space by $Φ_{1} (k_{x}, k_{y})$ [4]

\begin{array}{l} Φ_{1} (k_{x}, k_{y}) = \exp (- a k_{x}^{2}) \exp (i (k_{x}^{3} - 3 a^{2} k_{x} - i a^{3}) / 3) \\ _{}^{}_{}^{}_{}^{}_{}^{} \underset{}{} \times \exp (- a k_{y}^{2}) \exp (i (k_{y}^{3} - 3 a^{2} k_{y} - i a^{3}) / 3) . \end{array}

k_{x}, k_{y}

is the corresponding wave vector in x and y directions, respectively. Here we omit the higher-order terms of constant a, [i.e.,

\exp (i (- 3 a^{2} k - i a^{3}) /3)

], because a is positive and actually much less than 1. In Eq. (1) a is equal to 0.05 in our experiment and theoretical simulation. As a consequence,

Φ_{1} (k_{x}, k_{y})

is given approximately by

Φ_{2} (k_{x}, k_{y})

,

\begin{array}{l} Φ_{2} (k_{x}, k_{y}) = \exp (- a k_{x}^{2}) \exp (i k_{x}^{3} / 3) \times \exp (- a k_{y}^{2}) \exp (i k_{y}^{3} / 3) . \\ _{}^{}_{}^{}_{}^{}_{}^{} = \exp [- a (k_{x}^{2} + k_{y}^{2})] \exp [i (k_{x}^{3} + k_{y}^{3}) / 3] \end{array}

Following the discussion above, it demonstrates that the Fourier spectrum of the Airy beam is actually a Gaussian beam imposed by a cubic phase $f (k_{x}, k_{y}) = (k_{x}^{3} + k_{y}^{3}) / 3$ . Since the 2D lineal potential Schrödinger equation [Eq. (1)] is separable in an orthogonal Cartesian coordinate system, one can obtain explicit closed-form solutions [Eq. (2)], which is just the product of two orthogonal 1D accelerating Airy beams [see Eq. (2)]. Hence, the Airy beam forms a right-angle shape.

It is this which motivates our idea as to whether we can generate a quasi-Airy beam (acute or obtuse-angle Airy beam) by rotating sidelobes of an Airy beam. However, when the two 1D Airy beams are not mutually perpendicular, this inevitably produces components for non-orthogonal 1D Airy beams in orthogonal Cartesian coordinate system. Therefore the product of two non-orthogonal 1D accelerating Airy beams is not the solution of Eq. (1). Although we cannot obtain the analytic solution of quasi-Airy beams from Eq. (1), we can get their numerical solution of such Airy beams, since we can get any desirable shape and intensity distribution if we can generate the matching modulation phase.

Next, we introduce tunable disturbance factors $θ_{1}$ and $θ_{2}$ into Eq. (4) in $k_{x} - k_{y}$ space. We firstly discuss the rotation theory. By rotating coordinate vector $\vec{k_{x}}$ and $\vec{k_{y}}$ with angle $θ_{1}$ and $θ_{2}$ , respectively, we obtain the new vector $\vec{{k^{'}}_{x}}$ and $\vec{{k^{'}}_{y}}$ , which is expressed in original orthogonal $k_{x} - k_{y}$ space by

\begin{array}{l} {k^{'}}_{x} = k_{x} \cos θ_{1} - k_{y} \sin θ_{1}, \\ {k^{'}}_{y} = k_{x} \sin θ_{2} + k_{y} \cos θ_{2} . \end{array}

Where the range of

θ_{1}

and

θ_{2}

is 0 to

2 π

. As a result, the Fourier spectrum

Φ_{2} (k_{x}, k_{y})

become into a novel form

{Φ^{'}}_{2} (k_{x}, k_{y})

following the above discussed rotation theory

\begin{array}{l} {Φ^{'}}_{2} (k_{x}, k_{y}) = \exp [- a (^{k_{x} \cos θ_{1} - k_{y} \sin θ_{1}) 2} - a (^{k_{x} \sin θ_{2} + k_{y} \cos θ_{2}) 2}] \\ _{}^{}_{}^{}_{}^{}_{}^{} \overset{}{} \times \exp [i (^{k_{x} \cos θ_{1} - k_{y} \sin θ_{1}) 3} / 3 + i (^{k_{x} \sin θ_{2} + k_{y} \cos θ_{2}) 3} / 3] . \end{array}

Equation (6) demonstrates that two disturbance factors [

θ_{1}

and

θ_{2}

] are introduced into

Φ_{2} (k_{x}, k_{y})

to form a new expression

{Φ^{'}}_{2} (k_{x}, k_{y})

, and the Fourier transform of

{Φ^{'}}_{2} (k_{x}, k_{y})

will generate a new family of Airy beam, named quasi-Airy beams. The included angle

α

is formed by two sidelobes, which are no longer always equal to

90 °

.

Figure 1 demonstrates the intensity profiles of quasi-Airy beams at different factors. Their topological structures and shapes are distinctly different from those of an Airy beam. The quasi-Airy beams can be regarded as abnormal Airy beams, which are formed by rotating x-sidelobe and y-sidelobe around the z axis clockwise on the basis of the Airy beam. As demonstrated in Fig. 1, we can obtain arbitrarily acute or obtuse-angle Airy beam. It must be emphasized that although the quasi-Airy beams demonstrated in Fig. 1(c) and Fig. 1(h) have the similar topological structure (here, the same included angle $α =60 °$ ), the intensity profile is obviously different. Moreover, what is important is that their propagation trajectories are totally different.

Fig. 1 Intensity profiles of quasi-Airy beams at different conditions.

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As demonstrated Fig. 1(f), when the factor $θ_{1} = π /6$ and $θ_{2} =2 π /3$ (that means the included angle $α$ is equal to 0), the quasi-Airy beam turns into a tilted 1D Airy beam owing to ${Φ^{'}}_{2} (k_{x}, k_{y}) = \exp [- 2 a {(\frac{\sqrt{3}}{2} k_{x} - \frac{1}{2} k_{y})}^{2}] \exp [\frac{2 i}{3} {(\frac{\sqrt{3}}{2} k_{x} - \frac{1}{2} k_{y})}^{3}]$ , which is just the Fourier spectrum of a tilted 1D Airy beam. Particularly, the quasi-Airy beam will evolve into a Gaussian beam when the included angle $α$ is equal to $180 °$ [Fig. 1(g)], because in this case ${Φ^{'}}_{2} (k_{x}, k_{y})$ is equal to $\exp (- 2 a k_{y}^{2})$ , which is just a Gaussian beam, and the Fourier transform of the Gaussian beam is itself. In contrast with Airy beams, the topological structures of quasi-Airy beams can be freely controlled by tunable disturbance factors. Furthermore, quasi-Airy beams also present a new way to control parabolic trajectory (see Section 3).

3. Experiments and dynamics

In experiments and applications, the finite-energy Airy beam can be constructed by the Fourier transform function, which is a Gaussian beam modulated by a cubic phase [2]. Therefore, according to Eq. (6) the tunable phase function $f^{'} (k_{x}, k_{y})$ is expressed by in $k_{x} - k_{y}$ system

f^{'} (k_{x}, k_{y}) = {(k_{x} \cos θ_{1} - k_{y} \sin θ_{1})}^{3} / 3 + {(k_{x} \sin θ_{2} + k_{y} \cos θ_{2})}^{3} / 3.

Experimental setup is demonstrated in Fig. 2. A He-Ne laser (632nm) is employed to generate a polarized laser beam, which is collimated and expanded into a Gaussian beam with 8.6mm (FWHM), and the Gaussian beam is reflected by a computer-controlled spatial light modulator (SLM), which is imposed by the tunable phase pattern. The pixel size and resolution of SLM are 8um and 1920 × 1080 pixels, respectively. Next the phase-modulated and reflected Gaussian beam performs the Fourier transformation by a Fourier lens with the focal length of $f^{'} =300mm$ . Finally, the intensity profiles in the x-y plane are recorded by a CCD. The SLM and CCD are put on the front focal plane and back focal plane of the Fourier lens, respectively.

Fig. 2 Experimental setup.

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The experimental intensity profiles are demonstrated in Fig. 3. Figures 3(a1)–3(e1) are the tunable phase patterns, where the corresponding tunable factor $θ_{1}$ and $θ_{2}$ are given in the second row, respectively. The corresponding experimental results are depicted in Figs. 3(a2)-3(e2). Also the corresponding numerical results are shown in Figs. 3(a3)-3(e3). It is quite clear that the experimental results are consistent with the numerical results. All panels show the beams at z = 0 plane.

Fig. 3 Tunable phase patterns and quasi-Airy beams.

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It is well known that Airy beams accelerate freely during propagation. This property is reflected in the terms $x - {(z / 2)}^{2}$ and $y - {(z / 2)}^{2}$ in Eq. (2), which demonstrates that the beam propagates along a parabolic trajectory [4], it also shows that the propagation direction of the Airy beam is along the fixed $45 °$ axis in the x-y plane. It must be pointed out here that we do not present an analytical expression of quasi-Airy beams. However, as demonstrated in previous any research [2,3,19,20], the quasi-Airy beam with an arbitrary included angle $α$ can be synthesized in z = 0 plane

\begin{array}{l} ϕ (x, y, z =0) = \prod_{n = 1, 2} A i (s_{n}) \exp (a s_{n}), \\ s_{n} = (^{- 1) n - 1} x \cos (α / 2) / r_{0} - y \sin (α / 2) / r_{0} . \end{array}

Here, $α$ is the included angle formed by two sidelobes. The scope is 0 to $π$ , and $s_{n}$ denotes a dimensionless transverse coordinate x or y, $r_{0}$ is an arbitrary transverse scale.

As discussed previously [21,22], the curved propagation trajectory of Airy beam is a caustic ray, which is known as a hyperbolic umbilic. According to Eq. (8), Fig. 4 demonstrates the side view of the quasi-Airy beam at a different included angle of $α$ employing split-step Fourier method. Here $r_{0} = 15 u m$ and $λ = 632 n m$ . The insets show the corresponding intensity profiles in the x-y plane.

Fig. 4 Side view of quasi-Airy beam along y-z plane propagation.

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It is obvious that the quasi-Airy beam generates two propagation trajectories $Y_{1}$ and $Y_{2}$ during propagation, namely the main lobe and peak intensity. As shown in Fig. 4(a) and 4(c), the main lobe and the peak intensity propagates along their respective parabolic trajectory, however, the trajectories $Y_{1}$ and $Y_{2}$ coincide when $α =90 °$ [Fig. 4(b)] . It also demonstrates that the Airy beam can be regarded as a special case of a quasi-Airy beam when $α =90 °$ .

To better clarify the hyperbolic umbilic of the quasi-Airy beam, a corresponding 3D quasi-Airy beam is depicted in Fig. 5. The 3D quasi-Airy beam consists of two curved surfaces, addressed as “caustic 1” and “caustic 2,” which intersect at the z = 0 plane.

Fig. 5 Caustic surfaces.

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As depicted in Fig. 5(b), the two surfaces of the caustic perfectly overlap for the Airy beam during propagation along the z axis. Therefore, the caustic surface has only one intersection in the x = y plane. Consequently, it causes only one intensity maximum. However, those of quasi-Airy beam separate away from the z = 0 plane, resulting in two intensity maximums, (i.e., the main lobe $Y_{1}$ and peak intensity $Y_{2}$ [Fig. 5(a) and 5(c)]). Figure 5 also demonstrates that the distance between the two caustics surface increases with the increasing propagation distance z. We notice that caustic 1 is a cuspoid, while caustic 2 is smooth.

To better elaborate the propagation behavior of quasi-Airy beam, the asymptotically exact analysis employing uniform geometrical optics (UGO) [22] is performed. The approximate formula of the Airy function $A i (x)$ in the case of $x < 0$ is given

A i (x) \approx {(- π^{2} x)}^{- 1 / 4} \sin [\frac{2}{3} {(- x)}^{3 / 2} + \frac{π}{4}] .

Next, substitute Eq. (9) into Eq. (8), while allowing $a = 0$ , and $s_{1} = x \cos (α / 2) / r_{0} - y \sin (α / 2) / r_{0}$ , $s_{2} = - x \cos (α / 2) / r_{0} - y \sin (α / 2) / r_{0}$ . In this way, we can achieve a field distribution at $z = 0$ plane

\begin{array}{l} ϕ (x, y, z =0) = A_{0}^{+} \exp (i ψ_{0}^{+}) + A_{0}^{-} \exp (i ψ_{0}^{-}) \\ _{}^{}_{}^{}_{}^{}_{}^{}_{}^{} = ϕ_{0}^{+} (x, y) + ϕ_{0}^{-} (x, y), \end{array}

ψ_{0}^{\pm} = \mp \arcsin (\sin [(2 / 3) {(- s_{1})}^{3 / 2} + π / 4]) \times \sin [(2 / 3) {(- s_{2})}^{3 / 2} + π / 4],

A_{0}^{\pm} = \pm (i / 2) {(- π^{4} s_{1} s_{2})}^{- 1 / 4} .

As expressed by Eq. (10), the optical field at $z = 0$ plane consists of two components, $ϕ_{0}^{+}$ and $ϕ_{0}^{-}$ , which radiates rays, and whose caustics are depicted as

X = x + z \tan [\sin^{- 1} (\partial_{x} ψ_{0}^{\pm})],_{}^{} Y = y + z \tan [\sin^{- 1} (\partial_{y} ψ_{0}^{\pm})] .

With this, we can acquire the hyperbolic trajectories of quasi-Airy beam, which are described by the theory of catastrophe optics [23–25]. Under the paraxial approximation, the hyperbolic trajectories (i.e., the trajectories of two intensity maximums) are expressed by the ray-system Jacobian

Y_{1} = - \frac{λ^{2} \sin^{3} (α / 2)}{4 π^{2} r_{0}^{3}} z^{2},

Y_{2} = \frac{(1 - 3 \sin^{2} (α / 2)) \cos^{2} (α / 2)}{\sin (α / 2)} \frac{λ^{2}}{4 π^{2} r_{0}^{3}} z^{2} .

Where $Y_{1}$ and $Y_{2}$ represent the trajectory of the main lobe and peak intensity of the quasi-Airy beam, respectively. $λ$ denotes wavelength. Two parabolic trajectories are depicted with different included angles in Fig. 6 (here $r_{0} = 15 u m$ and $λ = 632 n m$ ).

Fig. 6 The tunable trajectory of the main lobe and peak intensity, The solid line and dashed line denotes $Y_{1}$ and $Y_{2}$ , respectively.

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It is clear that the main lobe (solid line) and the peak intensity (dashed line) propagate along their respective trajectories except in the case of $α =90 °$ . There, the $Y_{1}$ and $Y_{2}$ coincides perfectly (red line). It also indicates that one can readily control the intensity of the main lobe or peak intensity to propagate along an arbitrary trajectory by tuning the angle factors.

As demonstrated in Eq. (2), one can know that main lobe of the Airy beam also follows a parabolic trajectory in the $x - z$ and $y - z$ plane, which is reflected by the term $x - {(z / 2)}^{2}$ and $y - {(z / 2)}^{2}$ , respectively. Therefore, the propagation direction of the main lobe is always along the 45° axis in the $x - y$ plane. However, for an accelerating quasi-Airy beam, the parabolic trajectory is related to $x \cos θ_{1} - y \sin θ_{1} - {(z / 2)}^{2}$ and $x \sin θ_{2} + y \cos θ_{2} - {(z / 2)}^{2}$ according the rotation theory. As a result, the propagation direction projected in the x-y plane follows the following curve shown below due to the introduced angle factors,

y = \frac{\cos θ_{1} - \sin θ_{2}}{\sin θ_{1} + \cos θ_{2}} x .

The tunable propagation directions of the main lobe are shown in Fig. 7 under different angle factors. The insets show the corresponding values of ( $θ_{1}, θ_{2}$ ) and intensity profiles. Actually, we also verify that the propagation directions always move along the bisector of the included angle $α$ . As shown in Figs. 6 and 7, one cannot only readily tune the beam wavefront, but can also flexibly control quasi-Airy beam to propagate along an arbitrary parabolic trajectory and direction by applying tunable angle factors.

Fig. 7 Tunable propagation direction.

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To demonstrate the optical behavior of quasi-Airy beams, optical experiments are performed by employing the mentioned setup [see Fig. 2], where the CCD can be moved along the optical axis. Figures 8(a1)-8(d1) demonstrate the intensity profiles at the propagation plane of z = 0, 3, 6, 10cm when $θ_{1}$ and $θ_{2}$ is equal to $π /3$ and $7 π /6$ , respectively. It is obvious that acute wavefront develops gradually into two caustics (caustic 1 and caustic 2) during propagation. It demonstrates that the energy of the main lobe diffuses outwards, and therefore, caustic 1 becomes smoother. However, those with two sidelobes flows inward, and consequently, caustic 2 becomes sharper with increasing propagation distance. This is consistent with the results described in Fig. 5. Finally, the acute quasi-Airy beam cannot keep its original shape perfectly, but we notice that the energy of the main lobe is still strong in a certain range. The corresponding numerical results are shown in Figs. 8(a2)-8(d2), and in accordance with the experimental results.

Fig. 8 Experimental intensity profiles of quasi-Airy beams $θ_{1} = π /3$ and $θ_{2} =7 π /6$ at propagation planes z = (a1) 0, (b1) 3cm, (c1) 6cm, (d1) 10cm, respectively. (a2-d2) The corresponding numerical results. (a3-d3) The corresponding numerical results of Airy beam. (a4-d4) The corresponding numerical results of quasi-Airy beam when $α =95 °$ .

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Figures 8(a3)-8(d3) present the diffraction-free property of an Airy beam, where we know clearly that the Airy beam retains its original shape and remain unchanged during propagation. As discussed in Section 2, the two 1D Airy beams of the quasi-Airy beam are no longer orthogonal, and the two sidelobes are mutually coupled together. As a result, they are not separable in an orthogonal x-y system. From the point of view of the non-diffraction definition (even though quasi-Airy beams are not non-diffraction beams), we must emphasize the fact that such quasi-Airy beams actually still hold the diffraction-free ability to a certain extent. This ability depends on the included angle $α$ . The closer angle $α$ is to $90 °$ , the stronger the non-diffraction ability. The most obvious case is when the included angle $α$ is equal to $95 °$ , the quasi-Airy beam nearly possesses the same diffraction-free ability compared with the Airy beam, as shown in Figs. 8(a4)-8(d4), the shape is almost invariable during propagation.

As mentioned above, quasi-Airy beams still have diffraction-free ability to some degree compared with the Airy beam, so it can be deduced that the property of self-healing also imperfectly exists in quasi-Airy beams. Subsequently, we utilize a non-transparent obstacle to obstruct the main lobe part, and move CCD along the optical axis, and eventually, the intensity profiles are captured by CCD. Figures 9(a1)-9(d1) demonstrate the intensity profiles of quasi-Airy beams in the case of $θ_{1} =3 π /2$ and $θ_{2} = π /4$ at z = 0, 3, 6, 10cm plane, respectively. The corresponding experimental results are shown in Figs. 9(a2)-9(d2). Obviously, the blocked quasi-Airy beam still reconstructs the main lobe to a certain extent during propagation, although it is not perfect compared with the blocked Airy beam [Figs. 9(a3)-9(d3)]. Similar to the above discussion, if the closer the included angle $α$ is to $90 °$ , the stronger the ability of self-healing is. Compared to the Airy beam, a quasi-Airy beam with $α =95 °$ nearly reconstructs perfectly this shape [Figs. 9(a4)-9(d4)].

Fig. 9 Numerical intensity profiles of blocked quasi-Airy beams when $θ_{1} =3 π /2$ and $θ_{2} = π /4$ at propagation plane z = (a1) 0, (b1) 3cm, (c1) 6cm, (d1) 10cm, respectively. (a2)-(d2) The corresponding experimental results. (a3-d3) The corresponding numerical results of Airy beam. (a4-d4) The corresponding numerical results of quasi-Airy beam when $α =95 °$ .

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4. Conclusions

In conclusion, we have presented an accelerating quasi-Airy beam, which can be generated by a tunable phase pattern. Such quasi-Airy beams can freely move along a predesigned trajectory and direction during propagation. The advantage can facilitate the generated beam to avoid obstacle and transport a particle to a desired location, moreover, their topological structure can be readily manipulated. Compared to Airy beam, the quasi-Airy beams also exhibit the properties of non-diffraction, and are self-healing to some extent. As demonstrated in our simulations and experiments, the experimental results are in accord with the theoretical simulations. It is believed that the intriguing characteristics of the quasi-Airy beam could provide more degrees of freedom for atom trapping and cell manipulation.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 61377014).

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25. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, 1993).

Quasi-Airy beams along tunable propagation trajectories and directions

Abstract

1. Introduction

2. Theory

3. Experiments and dynamics

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Equations (16)

Optics Express