1. Introduction

Very recently a new “diffraction-free” optical beam, whose profile along one or two lateral axes is governed by the Airy function, has been derived and studied both theoretically and experimentally [1–3]. This finding is based on the mathematical equivalence between the scalar paraxial wave equation in variables x, z and Schrödinger equation in variables x, t and stems from an old but intriguing paper [4] by Sir Michael Berry and Nandor Balazs. Namely they predicted that if the wavefunction of a quantum mechanical particle is in the form of the Airy function, then its modulus squared (probability density) propagates in free space without distortion and, moreover, exhibits constant acceleration despite absence of any force.

They resolved the apparent paradox with noting that since the Airy function is not square integrable, the center of gravity of the packet – the expectation value of the particle position x – cannot be defined. However, in Ref. [1] a new confined Airy solution to the paraxial wave equation was found, which lacks this shortcoming. At the same time, as the authors of Ref. [1] write “…the Airy wave packet still exhibits its most exotic feature, i. e., its trend to freely accelerate. This characteristic is rather peculiar given the fact that it may occur in free space, e. g., in the absence of any index gradients…”

It should be noted that the term ‘acceleration’ is not related to time here but has been used to describe the apparent deviation of the Airy beam from straight forward propagation direction. Indeed, as far as the wave field considered in [1–3] is monochromatic, its intensity distribution is stationary, i. e., nothing in it is moving.

The motivation of the given work is to study pulsed versions of the Airy beam, which should directly reveal the lateral acceleration of flying short pulses. Certain sophisticated spatiotemporal wavepackets have been already considered in [1], which combine the lateral profile of Bessel-Gauss beams with the new Airy solution depending on the longitudinal and temporal variables. However, such wavepackets are not short-pulse versions of the Airy beam, since they exhibit the Airy profile and acceleration in the longitudinal direction [1].

As a matter of fact, any non-spreading Airy pulse becomes a member to the family of so-called localized waves – wideband pulses as generalizations of “diffraction-free” (quasi)monochromatic beams. Localized waves as a research subject has matured considerably over the last decade (see monograph [5] and, e. g., a review [6]), while it coped with several exotic features of new solutions to the wave equation.

2. Three types of Airy pulses

A general expression for the wave function of 2-D optical pulses possessing the Airy profile along one lateral axis (x) and propagating along axis z is given by a superposition of monochromatic constituents with wavenumbers k over a spectrum S(k)

where the factor exp(ikz) originates from the paraxial approximation ansatz and Φ is the scalar field envelope function of the finite-energy Airy beam [1–3]:

Here Ai(…) is the Airy function, the aperture parameter a is a small positive number so as to ensure containment of the infinite Airy function tail at negative values of x, generally wavenumber-dependent parameters x ₀ and z ₀ are the transversal and longitudinal characteristic lengths, respectively. The modulus of Φ has mirror symmetry with respect to the plane z=0.

However, the scales x ₀ and z ₀ are not arbitrary, the paraxial wave equation superimposes the relation z ₀=k x ₀ ² on them, which is an equivalent to one between the confocal parameter (twice the Rayleigh range) and the waist of the Gaussian beam. This relation brings the k-dependence into Φ(…). In order to make use of Eq. (1) it is not enough to specify the spectrum S(k), a k-dependence of either x ₀ or z ₀ has to be prescribed as well as it is done in the case of pulsed Gaussian beams [7–10].

In accordance with typical experimental situations, three different cases of the k-dependence have been pointed out by Sheppard [11]. A pulse with constant spectral distribution in the waist, i. e., with z ₀ proportional to the wavenumber, is termed as type I pulsed beam. Type II is the case when the spectral distribution is the same for all points across the aperture of the focusing optics and in the far field, then z ₀∝1/k, i. e., the confocal parameter is proportional to the wavelength. Finally, type III beam is an intermediate case when the confocal parameter is constant for all spectral components, which is the case when, e. g., the pulsed beam is generated in the open resonator of a mode-locked laser.

Having in mind experimental feasibility, we have chosen the following values of the parameters in Eq. (1)–(2) for numerical simulations. The containment parameter a=0.025 was taken so small that the field pattern of the monochromatic constituents of the pulse practically does not deviate from that of the “ideal” infinite-energy Airy beam [1, 2]. The scale parameter x ₀=0.1 mm for the central (carrier) frequency component of the pulse approximately equals to the HWHM of the first Airy peak in the initial cross section (focal) plane at z=0. The central wavenumber of a Gaussian spectrum was taken k ₀=10⁵ cm^-1, i. e., the corresponding wavelength 0.628µm falls into the red region. For the carrier frequency one finds from the relation z ₀=kx ₀ ² that z ₀(k ₀)=0.1 m.

Type I Airy pulse is the simplest in the sense that as the scale parameter x ₀ is the same for all the frequency components the transversal profile of the pulse in the initial plane z=0 should coincide with that of the monochromatic Airy beam (see [3], Fig. 1).

Indeed, the simulations in Fig. 1 and 2 are in accordance with such guess. The Airy pulse behaves as if flying around the corner – it deviates from straight propagation direction, whereas there is no cause of such transversal acceleration in the free space. The narrowband pulse (Fig. 1) preserves its transversal Airy profile in the course of propagation, while the wideband one changes it substantially during the acceleration stage.

 

Fig. 1. A sum of frames taken at instants ct=6 cm, ct=45 cm, and ct=84 cm of the animated plot (0.6 MB) of profiles of a type I narrowband Airy pulse which have been computed as modulus squared of the wavefunction given by Eqs. (1), (2). The plot can be also considered as a snapshot of three sequential pulses separated by an interval ct=39 cm. The full width at 1/e level of the spectrum is 2×10⁵ times smaller than the carrier frequency k ₀=10⁵ cm^-1. Correspondingly, the pulsewidth (intensity FWHM) is 1/3 ns. Other parameters are given in the text. Pseudocolor scale – from dark blue (zero level) up to red (peak value).[Media 1]

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Generally differences between type I, II and III pulsed beams become remarkable if the pulses contain only few cycles of the carrier, i. e., if the pulse’s spectral width becomes by its order of magnitude comparable to the carrier frequency. Therefore for Fig. 2 and for the subsequent simulations the full width at 1/e level of the spectrum was chosen 20% of the carrier frequency k ₀=10⁵ cm^-1. In accordance with the relation between a Gaussian curve and its Fourier image, the pulsewidth (intensity FWHM) is 8 fs. Such a wideband pulse is so short that in order to depict the modulus of its wavefunction, a co-propagating window with high resolution in the z-direction was used. Also, the peak value of the modulus was normalized to unity, so its decay in the course of propagation according to the exponent in Eq. (2) is not seen in the animations. The decay is not substantial – the peak value in the last (31^st) frame is approximately 3 times less than that in the 1^st frame in the case of all animations (for the taken value of a=0.025).

 

Fig. 2. The frame t=0 of the animated plot (1.4 MB) showing the modulus of the wavefunction of a type I wideband Airy pulse. Despite wavelengths of the monochromatic constituents of the pulse vary in a wide range the pulse Airy profile at z=0 is not smeared out, since the lateral scale x ₀ is identical for all constituents in the case of type I pulse. Intensity temporal FWHM is 8 fs. [Media 2]

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The profile of the type II pulse (Fig. 3) in the initial plane z=0 follows the Airy function curve in the region of a few main lobes and futher it exhibits letter-X-like branches or “wings” on the one side. It is remarkable that two-sided X-like profile is a characteristic feature of Bessel-X pulses and other optical localized waves [5, 6], which are ultrawideband superpositions of Bessel beams. Smearing out the oscillations of the Airy function and replacing them by the wings are due to the dependence of the lateral scale on the wavenumber: x ₀∝1/k or x ₀=(k ₀/k)×100µm in the case of values chosen by us. For our wideband pulse possessing 20% relative bandwidth and spectrally spanning from green to infrared (in the wavelength scale from 0.51 µm to 0.82 µm), x ₀ varies from 80 µm to 130 µm, which is enough to get the Airy oscillations of the pulse’s spectral constituents out of phase as soon as x<-0.5. Indeed, due to variety in values and sign of the Airy factor of the spectrum Eq. (2) within the range of the wavenumbers, e. g., at the point (x=-1, z=0) the Gaussian-shaped temporal spectrum acquires a sinusoidal modulation. The sinusoidal spectrum exactly corresponds to the double pulse recordable at this location when the “wings” fly through it according to Fig. 3.

 

Fig. 3. The frame t=0 of the animated plot (1.6 MB) of the modulus of the wavefunction of a type II wideband Airy pulse. The Airy profile is partially smeared out already at z=0 and replaced by a X-like branches, since the lateral scale x ₀ varies strongly – x ₀ is proportional to wavelength of a constituent of the type II pulse. Intensity temporal FWHM of the main lobe is 8 fs. [Media 3]

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In the case of the type III pulse the longitudinal scale parameter z ₀ is constant and therefore the frequency dependence of x ₀ is weaker: x ₀=(k ₀/k)^-1/2×100µm. Correspondingly, the initial pulse profile (Fig. 4) is something intermediate between that of the type I and type II pulse of the same bandwidth.

 

Fig. 4. The frame t=0 of the animated plot (1.2 MB) of the modulus of the wavefunction of a type III wideband Airy pulse. Intensity temporal FWHM of the main lobe is 8 fs. [Media 4]

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It is spectacular, however, that while in the last frames with large values of z the wing-like pattern of the type I and II pulse are of opposite manner, the type III pulse has no such pattern there. In spectral picture it is well understandable: since for the type III pulse the scale z ₀ is independent of k, according to Eq. (2) the spectrum Φ is also independent of k if x=0 and therefore along the propagation axis the Gaussian temporal profile of the pulse does not change and no doublets appear.

3. Discussion

The simulations apply not only for the chosen value of the carrier wavenumber, since actually the calculations use relative units and the plots are invariant under appropriate scalings. While the right-most high-value lobes of the three pulses obviously accelerate to the right in the course of propagation, does the pulse as a whole also “fly around the corner”? In other words, what is the behavior of the centroid (average lateral position) of the pulse when z increases? Besieris and Shaarawi have proved for the monochromatic finite-energy Airy field that the centroid does not depend on z at all [2]. Recently it was proven also experimentally that the center of gravity of the Airy beam does not accelerate [12]. The same must hold for the polychromatic pulses. Different behavior of the centroid and the peak is natural as generally for a distribution of a variable its mean value needn’t to coincide with the most probable value (of the lateral coordinate of photon – in the given case).

The right-most peak-value lobe in Fig. 3 exhibits propagation velocity very slightly exceeding c. This is not surprising because group velocity of wave packets in vacuum under specific circumstances may be superluminal [6, 13] (but not signalling velocity, of course). This is the case if the confocal parameter of a pulsed Gaussian beam is proportional to the wavelength [10, 7] – just like for our type II Airy pulse. This pulse is the best candidate for experimental realization as well, since by making use of the thin-lens Fourier transformation of a plane-wave pulse transmitted through a spatial modulator with cubic phase dependence along one lateral co-ordinate we get the type II dependence of x₀ on wavenumber. Experimental study of Airy pulses is the theme of our forthcoming publication.

Finally, it is instructive to note that an exact pulsed solution to the wave equation exists, which can be mathematically derived from the monochromatic approximate solution given by Eq. (2) through the following substitutions: k→2k, z→z-ct, and Φ→Φ·exp[ik(z+ct)]. This proposition is based on Sezginer’s finding [14] that any solution of the paraxial equation can be transformed in this way into a non-spreading wave of so-called focus wave mode type, which obeys the exact wave equation and due to the combined propagation variable z-ct preserves its shape and flies with the speed of light c. An expression of such wave has been derived in [15] for the case a=0, i. e., starting from the infinite-energy Airy beam.

4. Conclusion

We have derived and numerically simulated three types of Airy pulses. They differ in frequency dependences of their parameters and exhibit different shapes and propagation features. The highest-intensity lobes of the pulses exhibit a noticeable acceleration in the lateral direction but this observation does not mean as if the pulse’s center of gravity deviates from the straightly forward direction.