1 INTRODUCTION

A beam of electromagnetic radiation in which the amplitude of the electric field strength is precisely given by the Airy function does not undergo diffraction smearing during propagation but has infinite energy and thus cannot be experimentally implemented. The beam used in practice is the one in which the transverse distribution of the field at small distances from its axis is almost entirely defined by the Airy function but is limited in width. In the theoretical description of propagation of these beams it is usually taken that the electric field strength at the entrance to the medium is defined by the Airy function multiplied by an exponential factor with only linear (Airy beam) or linear and quadratic (Airy–Gauss beam) dependence of the argument on the transverse coordinate [1]. At rather large distances they almost do not suffer diffraction. Possibilities of controlling the amplitude, trajectory, width, and degree of curvature are well studied for the Airy beam [2–5], which is widely used in problems of optical manipulation of particles [6–12]. Airy–Gauss beams are not so popular. By now, it is mainly some features of their propagation in various medial that have been investigated [1, 13, 14], while a possibility of employing them in problems of optical manipulation has not even been discussed (in recent comprehensive reviews [11, 12] on laser manipulation of particles there is not a single word about Airy–Gauss beams). Below it will be shown that Airy–Gauss beams not only better approximate propagation of the “ideal Airy beam” with the transverse distribution of the electric field strength given by the Airy function alone but also are sometimes more preferable for optical manipulation of particles. The latter arises from the additional possibility of controlling the form of the transverse spatial distribution of their intensity in any plane perpendicular to the propagation axis.

2 OPTIMIZATION OF APPROXIMATION OF AN “IDEAL AIRY BEAM” BY THE AIRY–GAUSS BEAM

Let in the plane ζ = z/zR = 0, where \({{z}_{{\text{R}}}} = {{kr_{0}^{2}} \mathord{\left/ {\vphantom {{kr_{0}^{2}} 2}} \right. \kern-0em} 2}\) is the Rayleigh length defined by the smallest characteristic transverse scale r0 of the beam propagating along the ζ axis, and k is the wave number, the amplitude of the electric field strength normalized to the maximum value be given by the formula

$$\begin{gathered} u(\xi ,0) = \sqrt {\frac{{1 + i{{\zeta }_{{\text{f}}}}{{m}^{2}}}}{{1 + \zeta _{{\text{f}}}^{2}{{m}^{4}}}}} \operatorname{Ai} (\xi + d) \\ \times \,\,\exp \left[ {i{{S}_{1}}(\xi + d) + \frac{{iS_{1}^{3}}}{3} - \frac{{{{\xi }^{2}}{{m}^{2}}(1 + i{{\zeta }_{{\text{f}}}}{{m}^{2}})}}{{1 + \zeta _{{\text{f}}}^{2}{{m}^{4}}}}} \right], \\ \end{gathered} $$
(1)

where Ai(ξ + d) is the Airy function, d is the initial shift of its argument along the ξ axis, S1 = ν – ia, ν = θ/γ is the parameter characterizing the launch angle θ to the ζ axis of the beam with the limiting factor a > 0, and γ = (kr0)–1 is the small parameter of the paraxial approximation [15–19]. The beam waist associated with the Gaussian exponential in (1) is located in the plane ζ = ζf, and its size is larger than r0 by a factor of 1/m (m < 1). In a similar way as in [1], we obtain the following formula for the normalized strength amplitude of the electric field u(ξ,ζ) at its diffraction:

$$\begin{gathered} u(\xi ,\zeta ) = \sqrt K \operatorname{Ai} \left[ {Q(\xi ,\zeta )} \right] \\ \times \,\,\exp \left[ {i\left( {{{S}_{1}} + \frac{{J\zeta }}{4}} \right)Q(\xi ,\zeta ) + \frac{{i{{{\left( {{{S}_{1}} + {{J\zeta } \mathord{\left/ {\vphantom {{J\zeta } 4}} \right. \kern-0em} 4}} \right)}}^{3}}}}{3} - {{\xi }^{2}}{{m}^{2}}K} \right]. \\ \end{gathered} $$
(2)

Here \(Q(\xi ,\zeta ) = J\left[ {\xi - \zeta \left( {{{{{S}_{1}}} \mathord{\left/ {\vphantom {{{{S}_{1}}} 2}} \right. \kern-0em} 2} + {{J\zeta } \mathord{\left/ {\vphantom {{J\zeta } {16}}} \right. \kern-0em} {16}}} \right)} \right] + d\), \(J = (1 - i{{\zeta }_{{\text{f}}}}{{m}^{2}})K\), and

$$K = \frac{{1 - i\left( {\zeta - {{\zeta }_{{\text{f}}}}} \right){{m}^{2}}}}{{1 + {{{\left( {\zeta - {{\zeta }_{{\text{f}}}}} \right)}}^{2}}{{m}^{4}}}}.$$
(3)

At m = 0 formulas (1) and (2) describe the Airy beam. Further we will separate the linear and the quadratic contribution in the exponent for the clear understanding of their influence on the characteristics of the propagating beam, and we will set a = 0 for the Airy–Gauss beam. The beams will be compared at a = m, which ensure identical widths of both exponentials in (1) and (2), and at identical shifts d.

When the launch angle v changes, the function |u(ξ, ζ)|2 does not change its form in the plane ζ = const but shifts along the trajectory ξ = ξtr(ζ,ζf), the equation of which is defined by equality to zero of the real part of the Airy function argument [3] in (2). The trajectory of the Airy–Gauss beam depends on ζf and is much more complicated than the parabolic trajectory of the Airy beam [3]. The simplest form of ξtr(ζ,ζf) is at ζf = 0:

$$\begin{gathered} {{\xi }_{{{\text{tr}}}}}(\zeta ,0) = {{\left( {1 + {{\zeta }^{2}}{{m}^{4}}} \right)}^{{ - 1}}} \\ \times \,\,\left[ {\xi - \frac{{\zeta \nu }}{2} - \frac{{{{\zeta }^{2}}\left( {1 - {{\zeta }^{2}}{{m}^{4}}} \right)}}{{16\left( {1 + {{\zeta }^{2}}{{m}^{4}}} \right)}}} \right] - d. \\ \end{gathered} $$
(4)

Numerical analysis of (2) has shown that the normalized intensity |u(ξ,ζ)|2 of the propagating wave is rather sensitive to variation in the shift d, which opens up the possibility of effectively controlling the beam characteristics that are necessary for solving problems of optical manipulation of particles. At the fixed ν and m it is found that there exists an optimal d that ensures the maximum of |u(ξ,ζ)|2 in the plane ζ = const; its value dopt(ζ) depends on ζf but almost does not change under the angle θ variations. For example, in the plane ζ = 4 (this value of the coordinate correlates with the experimental conditions [9]), when ζf varies from zero to four, dopt hardly changes at m = 0.1 but decreases from –1.5 to –1.7 at m = 0.3. Approximation of the “ideal Airy beam” (m = a = 0) by the Airy–Gauss beam is better than by the Airy beam, including at d = 0 for 0 ≤ ζf ≤ 4 and at coinciding m and a (0 ≤ m ≤ 0.5, 0 ≤ a ≤ 0.5). The choice of the value dopt for d makes his approximation more accurate. For example, at m = 0.1, ζ = 0, and d = 0 the principal, first, and second maxima of the Airy–Gauss beam intensity are 0.98, 0.81, and 0.63 of the corresponding maxima of the ideal Airy beam. At the optimal shift the principal and first maxima are reproduced almost exactly, and next maxima are also reproduced much better (Fig. 1). At d = dopt the Gaussian exponential covers both maxima of the Airy function each way from the point where the maximum of the Gaussian function is reached. In this case, the attenuation of the principal and the first side maximum of the Airy function is almost identical, unlike the case involving influence of the exponential with the linear decay index. It not only attenuates the first side maximum greater than the principal one but also appreciably distorts the principal maximum of the Airy function, suppressing it in the region of ξ < 0 and increasing it when ξ > 0. These distortions will increase at the diffraction of the beam because of its propagation along a parabolic trajectory [3] towards positive ξ. It is this behavior of the exponential with a linear argument that does not allow independently “shifting” the Airy function, because Q(ξ,ζ) enters into both these functions. Obviously, the exponential limiting the intensity of the ideal Airy beam should involve a quadratic rather than a linear term.

Fig. 1.
figure 1

(Color online) Dependence of the function |u(ξ,0)|2 at ν = 0, d = –2 for the Airy–Gauss beam (m = 0.1, ζf = 4, red curve), Airy beam (a = 0.1, blue curve), and square of the Airy function (а = m = 0, dashed curve).

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The best approximation by the Airy–Gauss beam applied to the normalized electric field strength given by the Airy function ensures an increase in its diffraction-free propagation length as compared to the Airy beam. This is clearly seen in Fig. 2, where colors show the intensity difference between the Airy–Gauss beam (m = 0.1) and the Airy beam (a = 0.1) at ζf = ζ, ν =0, and dopt = –2. It is positive in the regions of plane ξζ corresponding to the intensity maxima and negative in the regions between the maxima of the beams; the latter means that scattering of the Airy beam is larger than scattering of the Airy–Gauss beam. The increase in the length of the diffraction-free Airy–Gauss beam propagation is also similarly manifested at ζf = 0.

Fig. 2.
figure 2

(Color online) Differences of the normalized intensities of the Airy–Gauss beam (m = 0.1 and ζf = ζ) and the Airy beam (a = 0.1) at ν = 0 and d = –2.

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3 AIRY–GAUSS BEAM CHARACTERISTICS SIGNIFICANT FOR OPTICAL MANIPULATION PROBLEMS

Formulas (1) and (2), which specify 2D Airy–Gauss and Airy beams, make it easy to write all necessary quantities that define forces and torques in problems of optical manipulation of micro- and nanoparticles using 2D and 3D Airy–Gauss beams. The slowly varying dimensionless amplitude of the 3D Airy–Gauss beam normalized to the square root of the beam intensity has the form U(ξ,η,ζ) = u(ξ,ζ) u(η,ζ); for the beams symmetric about ξ and η = y/r0, the function u(η,ζ) is obtained by replacing ξ with η in the formula for u(ξ,ζ). In the dipole approximation, the forces acting on micro- and nanoparticles depend on the time-average energy gradient (gradient force) and the momentum density vector of the electromagnetic field (scattering force), and in this case, the spin angular momentum density governs rotation of particles [18–22]. Projections of the densities of the orbital (canonical) momentum and the spin part of the angular momentum on the ζ axis are proportional to the normalized intensity |U(ξ,η,ζ)|2 or |u(ξ,ζ)|2 for 3D or 2D beams respectively [17–19]. As was shown earlier, in the case of using Airy–Gauss beams, these quantities can not only be much larger than for the Airy beam but also nonmonotonically vary with controlling parameters. In the case of using Airy–Gauss beams, the longitudinal component of the gradient force proportional to ∂|u(ξ,ζ)|2/∂ζ is not only larger than in the case of using Airy beams (Fig. 3) but also appreciably increases when d = dopt is chosen. For a 3D beam, this increase will be even larger because of increasing |u(η,ζ)|2. This means that in experiments on motion of particles along trajectories similar to those conducted in [6] with the Airy beam the use of the Airy–Gauss beam will open up new possibilities. The beam launch angle ν > 0 slightly increases the oscillation amplitude of the function ∂|u(ξ,ζ)|2/∂ζ, but at ν < 0 oscillations almost disappear, and the gradient decreases by about an order of magnitude as compared to the case of ν = 0.

Fig. 3.
figure 3

(Color online) Dependence of ∂|u(ξ,ζ)|2/∂ζ on ξ at ν = 0, ζ = 4, and d = –2 for the Airy–Gauss beam (m = 0.1, ζf = 4, red curve) and the Airy beam (a = 0.1, blue curve).

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In experiments with counterpropagating beams, the longitudinal components of the gradient and scattering forces are zero [8, 9, 22]. There may also be no total spin angular momentum, and thus the transverse components of these forces and of the spin angular momentum, which are of the first order of smallness in parameter γ, may become defining ones. These quantities are proportional to Re(u*∂ξ,ηu) and Im(u*∂ξ,ηu) [17–19]. A typical dependence of Im(u*∂ξu) on ξ and ν at fixed m, ζ, and d is shown in Fig. 4a. An increase in ν qualitatively changes the distribution of Im(u*∂ξu) in the plane ξν. This is especially distinct in the region of negative ν ≈ –1. The absolute value of Im(u*∂ξu) decreases by almost two orders of magnitude and becomes negligibly small, as is also the case for the Airy beam. At small m, when ζf increases from zero to four, the extrema of Im(u*∂ξu) shift leftwards, as do the extrema of Re(u*∂ξu) (Fig. 4b). Values of local extrema of Re(u*∂ξu) and average distances between the points where they are reached almost do not change with varying ν, and only the nonzero part of the function is “displaced” along the ξ axis in accordance with the shape of the trajectory. In this case, Re(u*∂ξu) defining the transverse projections of the spin momentum is larger in the Airy–Gauss beam than in the Airy beam. This should be borne in mind, since the force determined by the transverse component of the spin constituent of the momentum has an azimuthal direction and causes rotation of particles in the Gaussian beam [22] and the Airy beam [20]. If the transverse gradient force is insufficient, particles can be ejected from the trap [22].

Fig. 4.
figure 4

(Color online) Values of Im(u*∂ξu) as a function of the variables ξ and ν at m = 0.1, ζf = ζ = 4, and d = –2 (a) and of the Re(u*∂ξu) dependence on ξ at ν = 0, ζ = 4, and d = –2 for the Airy beam (a = 0.1, blue curve) and the Airy–Gauss beam at m = 0.1, and at ζf = 0 (solid red line) and ζf = 4 (dashed red line) (b).

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4 CONCLUSIONS

Diffraction of the Airy–Gauss beam has been analyzed in the paraxial approximation for the case where its waist associated with the Gaussian exponential is in an arbitrarily located plane perpendicular to the propagation axis. When the waist is in the observation plane, additional beam focusing is ensured, and the diffraction-free propagation length increases. It is shown that according to the quality criteria for the approximation of the Airy function, the characteristic diffraction-free propagation length, beam intensity, and its derivatives at different points of space, the Airy–Gauss beams are superior to the Airy beams, which are often used in experiments and are easier to be theoretically described. In the Airy–Gauss beams, the intensity and its derivatives determining, for example, the longitudinal and transverse components of the energy gradient and densities of the orbital and spin components of the momentum and the spin angular momentum are not only larger in magnitude but also better controlled in an experiment by shifting the argument of the Airy function and by the launch angle, the width, and the location of the beam waist. All this allows Airy–Gauss beams to be considered as more promising than Airy beams for solving problems of optical manipulation of micro- and nanoparticles.