1. Introduction

The characteristics of accelerating Airy beams are a subject of intense current interest because they represent a novel class of non diffracting field [1–3] and because of their tremendous application potential in fields such as optical manipulation, filamentation and nonlinear optics [4–9]. Since their first experimental observation in 2007, there have been extensive investigations into their fundamental propagation properties [10–18] as well as recent demonstrations of their importance in even wider fields of physics such as electron acceleration and plasmonics [19,20]. A central physical feature of Airy beams is that their accelerating trajectory is an example of a diffraction catastrophe, a localized region of concentrated intensity associated with the envelope caustic to a family of tangential geometrical rays. Although this was pointed out in the early work of Berry and Balazs [1], it is only recently that explicit analysis of the diffraction integral for an accelerating beam has shown how this concept actually allows the engineering of a much wider class of paraxial acceleration profiles through the design of an appropriate generating spatial phase mask [21]. This recent work is also significant in generalizing earlier proposals where the specific form of a serpentine trajectory was proposed using Fourier plane synthesis [22].

In fact, the link between an accelerating beam and a family of tangential rays to a caustic surface is an example of the geometrical Legendre transform, previously applied in a closely related context to determine caustics from reflecting surfaces and vice-versa [23–25]. In this paper, we show how these simple geometrical arguments in fact yield general phase profile design criteria for accelerating beams, even in the non-paraxial regime. We present both numerical and analytical results for a number of illustrative examples, and we report experiments that have synthesized parabolic, quartic and logarithmic accelerating profiles. We demonstrate compatibility with applications by generating these tailored beams using broadband 100 fs pulses and imaging the acceleration trajectories down to sub-10 μm dimensions. Significantly, the ability to image the generated phase profile with our setup allows us to position the accelerating trajectory at an arbitrary position within a dielectric sample, allowing us to use femtosecond accelerating beam to write a curved zone of refractive index modification in glass. We also show how a direct phase mask associated with a given caustic can be applied with both reflection and rotational symmetry on an incident beam to generate novel two- and three-dimensional volumetric accelerating beams bounded by a caustic surface of localized high intensity or superimposed with an optical spiral structure. We also present results that show how the accelerating boundaries of the volumetric profile converge in both two and three dimensions to a regime of linear abrupt autofocussing with transverse beam diameters less than 5 μm [6,8].

2. Acceleration profile engineering via caustics

Although Airy beams are often generated by imposing a cubic phase in the Fourier plane, they have also been synthesized directly on an incident Gaussian beam by encoding a 3/2 spatial phase function expressed in terms of the spatial transverse coordinate [26]. Deriving this phase condition uses geometrical arguments that relate the parabolic caustic trajectory of the Airy beam to a corresponding direct-space spatial phase, and it is this approach that we generalize here to arbitrary acceleration trajectories.

The approach is illustrated in Fig. 1(a) . The desired acceleration trajectory is defined by the curve c(z) as shown, and we wish to determine the corresponding spatial phase function ϕ(y) at the plane z = 0 that will generate this curve as a caustic. A caustic is defined as an envelope to a family of tangents such that each point y at the plane z = 0 can be functionally related to a point on the caustic via a tangent of slope θ where tanθ = c’(z) = dc(z)/dz. Since the tangent can be parameterized in terms of y using c’(z) = [c(z) – y]/z, we can determine the desired phase function by integrating the phase derivative condition:

 

Fig. 1 (a) Geometrical construction of a generating phase profile from the properties of an arbitrary desired caustic. The key concepts are (i) to identify the phase derivative dϕ/dy along the plane of the phase mask with the tangent slope to a point on the caustic and (ii) to use the parameterization of the slope c’(z) in terms of y (the Legendre transform) to allow integration to yield ϕ(y). The figure is drawn for the exact non paraxial case. In the paraxial case the approximation sinθ ≈ tanθ ≈θ allows some explicit results to be obtained as described in the text. (b) shows results of this approach comparing target parabolic caustics (white dashed line) with the propagated field using a Gaussian beam to which is applied the calculated phase profile (false color image). The approach is shown to work in both the paraxial (left) and non-paraxial (right) regimes with maximum ray angles to the caustic of 11° and 60° respectively.

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When the analysis in the preceding discussion is simplified in the paraxial approximation, we obtain the same results as those derived in Ref. 21 using analysis based on the first-order expansion of the diffraction integral. Working within the paraxial regime is also convenient as it allows the utility of the technique to be illustrated through several simple analytic examples where the caustic extrema are located at the desired phase plane. Specifically, (with y = |y| for compactness of notation), Table 1 below gives several particular results of interest in what follows.

Table 1. In the Paraxial Approximation, this Table gives the Calculated Phase for Desired Acceleration Profiles as Shown.

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3. Experimental setup and sheet caustics

The experimental setup we have implemented is shown in Fig. 2 . 100 fs pulses at 800 nm with Gaussian spatial intensity profile from a regeneratively-amplified Ti:Sapphire laser system (Spectra Physics Spitfire) are incident via a beam expander upon a spatial light modulator (SLM, Hamamatsu PPM X8627) to which the desired caustic-generating phase profile is applied. More specific technical details of our experimental system as applied to other classes of non-diffracting beam are given in Ref. 27. The phase function is applied over a beam size of 2 cm illuminating the full extent of the SLM. With the application of the phase profile directly, the accelerating beam is generated immediately after the SLM, but we further use a reduction telescope to image the accelerating beam down to micron-scale transverse dimensions in the vicinity of the caustic intensity maximum. The telescope is built with two converging lenses. We have used demagnification factors from ×10 to ×280 in our experiments. For a demagnification factor of 10, we used f₁=1000 mm and f₂=100 mm. Note that imaging in a 4f configuration (see Fig. 2) preserves the acceleration profile imposed by the phase mask at the SLM and allows a wide range of spatial frequencies to be used in constructing the desired acceleration profile. Also note that with the 100 fs pulses in our experiments (10 nm spectral FWHM), the effect of material dispersion in our system has negligible effect on the characteristics of the generated caustic profile. Our experiments therefore represent an important confirmation that arbitrary acceleration profile design technique [21] can be applied to applications involving femtosecond pulses and micron-dimension spotsizes.

 

Fig. 2 Schematic of our experimental setup. The spatial profile of incident 100 fs pulses is shaped with an appropriately-designed phase mask to directly generate a desired convex acceleration trajectory. A beam reduction telescope images the profile to micron-size dimensions where its evolution can be characterized, or where it can be applied to material processing. The figure illustrates in particular the creation of a two dimensional caustic sheet.

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In our experiments, we first applied the phase profile identically along each column of the SLM to generate two dimensional accelerating beams as illustrated in the schematic of Fig. 2. The principal lobe in this case takes the form of an “accelerating sheet”. We performed experiments in this configuration using phase masks corresponding to parabolic, quartic and logarithmic caustics as discussed above. With demagnification of ×10, we characterized the intensity profile as a function of distance from the image plane of the SLM after the telescope, and these results are shown in Fig. 3 .

 

Fig. 3 Comparing numerical beam propagation as described in the text with experimental measurements for (a) parabolic; (b) quartic; (c) logarithmic beam profiles as indicated. The position z=0 corresponds to the image plane of the SLM after the ×10 demagnification system used in this configuration.

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Before implementing the experiments, we used numerical solution of the full wave equation to check that the geometrically-calculated phase profile for each caustic yielded the desired behavior under realistic conditions that included a Gaussian intensity profile of the incident field and non-paraxial propagation. These results are also shown in Fig. 3. The results in Fig. 3 show very good agreement between the numerically calculated acceleration profiles and those measured in experiment. The geometrical target caustic is shown as the white dashed curve in each figure and it is clear that the numerical propagation results show the target caustic should be attained in all cases. This is expected from the results in Fig. 1(b) for the case of the parabolic (Airy) trajectory, but it is of course important to confirm it for arbitrary caustics.

The experimental results also shown in the figure follow the target caustic very closely but there is some deviation from the ideal case in the region close to the applied phase plane (plane z=0). Additional modeling of non-ideal propagation scenarios have clarified that this arises not because of any intrinsic limitation to the technique, but rather because the finite resolution of our SLM. More quantitative comparisons between the experimental and numerically-propagated beams are presented in Fig. 4 . For each case, we compare the numerical and measured intensity profiles transverse to the direction of propagation in the vicinity of the caustic extrema. We see very good agreement in both the width of the intensity maxima as well as the periodicity of the fringes on the exterior region of the caustic. Significantly, irrespective of the target caustic, the central intensity lobe exhibits the qualitative characteristics of the primary Airy lobe, providing additional confirmation of the analysis in Ref. 21 showing that the Airy function represents a universal feature of the intensity distributions of any acceleration profile in the paraxial regime.

 

Fig. 4 Line profile comparisons in the vicinity of the trajectory minima shown in Fig. 3. The solid black lines are the numerically propagated beam intensity profiles. The red lines with markers show the experimental measurements.

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4. Volume caustics and extension to sub-10 μm dimensions

The results above were obtained using ×10 demagnification in the imaging system, and yield primary intensity lobes of transverse widths ~20-30 μm as shown in Fig. 4. Significantly, the size of this intensity lobe can be readily modified simply by changing the imaging scale factor, which is important for tailoring the generation of accelerating beams to specific applications. We have generated results similar to those in Figs. 3 and 4 using a modified setup with higher demagnification factor of ×120, which has allowed us to attain ~5 μm transverse lobe dimensions. In fact, we have also combined this higher regime of demagnification with the application of reflection and rotation symmetry to the calculated phase mask, and we now present our results obtained in this regime.

Figure 5 shows experimental results where the calculated phase function associated with (a) parabolic and (b) quartic acceleration trajectories is truncated and reflected about the centre of the SLM to generate two sheet caustics that diverge at the plane of the imaged SLM before reconverging when the caustics accelerate beyond their extremal points. The phase reflection in is implemented using ϕ(y)=φ_c(y)∙H(y)+φ_c(-y)∙H(-y) where φ_c(y) is the calculated single caustic phase and H(y) is the Heaviside step function. The applied phase functions are shown in the left panels of the figure. The caustic-bounded beam profiles obtained experimentally are shown in the central panels, and were measured after ×120 demagnification. We also show here as the white dashed line the target acceleration profile from which the phase function was calculated. As with the results in Fig. 4, the experimental measurements confirm the ability of this technique to generate the target acceleration trajectories. The figure also shows line profiles at different points along the direction of propagation as indicated. The profiles at the points A illustrate the near zero intensity in the central region and the Airy fringes at the exterior of the beam. At the points of convergence labeled B in the figure, our results show signatures of the abrupt autofocussing phenomena as reported in Refs. 6-8 where the interference between the recombining caustics leads to a local intensity enhancement. Our two dimensional results here are associated with an increase of an order of magnitude in intensity between the peaks in point A and B labeled in both figures.

 

Fig. 5 Characterisation of a volume sheet caustic bounded by (a) two parabolic acceleration trajectories and (b) two quartic acceleration trajectories. In each case, the left panel shows the applied phase profile, the middle panel the intensity distribution measured from the SLM image plane (z = 0) after demagnification, and the right panels show the extracted intensity line profiles at the points A and B indicated. The white dashed lines in the centre panel shows the target acceleration trajectories. The intensity line profiles are plotted in arbitrary units but there is an order of magnitude increase in peak intensity from points A to B in both cases because of the autofocussing due to caustic recombination.

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The phase masks calculated for parabolic and quartic caustics can be readily combined with additional rotational phase terms in order to generate novel classes of optical beams in three dimensions. These results are shown in Fig. 6 and 7 . In Fig. 6 we show two different types of rotational phase applied to a parabolic caustic (Airy beam). In Fig. 6(a) (left panel) we apply simple rotational symmetry to the calculated caustic phase ϕ_c(r) as a function of radial distance r from the centre of the SLM as shown. As with the two dimensional results shown above, we measure the intensity distribution beyond the image plane of the SLM but in order to highlight the three-dimensional nature of the imaged field and the clear volume boundary associated with the caustic, we use a tomographic representation in the figure with partial transparency. The left subfigure is a transparent slice through the beam centre whilst the perspective selected in the right subfigure illustrates clearly the three dimensional nature of the caustic volume. The particular case chosen for illustration here is that of a rotationally-symmetric quartic caustic but similar results are obtained for other caustic forms and we present additional results for both quartic and parabolic caustics in Fig. 7 below. The demagnification factor here is again ×120 so that we obtain intensity lobes of sub-10 μm dimension as we shall see.

 

Fig. 6 Combining an engineered acceleration trajectory with (a) rotational symmetry and (b) an imposed spiral structure. The left panels show the applied phase profiles; the right panels show tomographic representations of the shaped fields in both cases as discussed in the text.

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Fig. 7 Intensity slice of the center of rotationally-symmetric beams bounded by (a) quartic and (b) parabolic caustic accelerating beams. The intensity line profiles shown to the right in each case correspond to the intensity maximum of the focal region transverse to the propagation direction (top) and along the propagation direction (bottom).

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In Fig. 6(b) the applied rotational phase is more complex. Specifically, we imprint the caustic phase within a binary annular spiral phase mask in order to show the compatibility of arbitrary acceleration beam engineering with techniques commonly used to synthesize optical vortex structures [28]. In particular, we apply a phase ϕ(r,θ) = φ _c (r) B(r,θ) where φ _c (r) is a rotationally symmetric quartic phase as in Fig. 6(a), and Β(r,θ) = rect[(r - aθ)/Δr] describes the binary spiral function. Here, θ varies between 0 and n2π and the parameters are a = r _max/2πn where n is the number of spiral turns of zeroing width Δr, and r _max is the maximal exterior radius of the spiral. In this case, the spiralling nature of the caustic surface means that tomographic visualisation in a plane as in the centre panel is not effective in illustrating the nature of the accelerating field, but this is shown very clearly with the perspective chosen in the right panel of Fig. 6(b).

We return now to the rotationally symmetric case of Fig. 6(a) in order to analyse in more detail the autofocussing phenomenon observed at the convergence point of the caustic boundaries. We note at this point that this topic is a subject of much current interest, with recent work in refs [6,7] studying this effect theoretically and experimentally for the specific case of circular Airy beams. Related work has also shown that chirped rings propagate following a caustic surface which collapse is the point of self-focus [8]. Our results here showing autofocussing with a polynomial trajectory are highly complementary to these studies, showing the generality of the autofocussing effect to a wider range of caustic profiles. These results suggest that the direct design of the caustic trajectory can extend the regime over which autofocussing caustic collapse can be observed, and indeed provide an additional engineering approach to the design of specific (autofocussing) focal characteristics.

In this context, we stress that we have observed autofocussing for a wide range of polynomial profiles, suggesting the universality of this effect as an intrinsic feature of the convergence of caustic accelerating beams. Moreover, these results are significant in showing that the autofocussing effect can be observed with direct spatial phase encoding in addition to the techniques of Fourier optics employed in Ref. 7. To analyse this in more detail, Fig. 7 shows line profile intensity slices of the measured field in the plane of maximum intensity along the direction of convergence towards the focus. The figure shows results for rotationally symmetric beams of (a) parabolic and (b) quartic caustic forms. The transverse dimension in the focal region is ~2.5 μm at the quartic beam focus and ~5 μm at the parabolic beam focus, with this difference arising because of the steeper convergence angles for the accelerating beam components arriving at the focal point for the quartic case. The quartic beam also achieves its focus more abruptly than the parabolic beam along the direction of propagation.

5. Applications to femtosecond micromachining

The ability to generate accelerating trajectories as described above naturally leads to the suggestion of important applications in material processing. To this end, we report here proof-of-principle results where femtosecond accelerating beams synthesized using an appropriate phase mask are used to write a region of curved refractive index modification in glass. For these experiments, the SLM was de-magnified by a factor 280, a similar scaling factor as that achieved in reference with non-diffracting Bessel beams [27]. We used parabolic beams in these experiments, and applied the phase profile along the two orthogonal directions of the SLM to provide crossing sheet caustics with an enhanced intensity at the intersection of the main intensity lobes. The measured intensity profile for this case is shown in Fig. 8(a) . With this beam under conditions of single shot illumination and at a pulse energy of ~100 μJ, we were able to write a curved region of refractive index modification, and Fig. 8(b) shows a differential interference contrast microscopy image of the curved index modified region of the glass. The dotted lines are parabolic segments of the beam trajectory superimposed on the image to illustrate how the index modification follows the expected accelerating path of the beam. We note here that the intrinsic structural stability of the caustic profile [1–4] means that, although the glass-air interface introduces a quantitative modification to the characteristics of the accelerating profile, the intensity localization associated with the caustic is preserved. Of course in other contexts this property is well-known as the “self-healing” property of non-diffracting beams. For the purposes of material processing, this is an extremely important characteristic, as it allows high quality structures to be machined under a wide variety of conditions as previously demonstrated with Bessel beams [27]. For completeness we note that for the paraxial regime, the quantitative change to the parabolic trajectory in the glass (refer to the notation in Table 1) can be readily calculated as c(z)=(a/n)(z-z₀)², where n is the index of refraction of glass and z₀ is the extremum of the parabola. More generally, to obtain a caustic c(z) in a medium of refractive index n, the phase to be applied at the interface requires simply an n-fold increase in the phase required to produce the same caustic trajectory in air.

 

Fig. 8 Results showing application of our setup to femtosecond waveguide writing. (a) shows a slice through a double Airy profile synthesized as described in the text. (b) shows the curved region of refractive index modification.

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6. Conclusions

The technique and results presented here are an important confirmation of the ability of direct spatial phase encoding and imaging to generate arbitrary acceleration profiles down to sub-10 μm dimensions, and to combine this beam engineering technique with straightforward approaches to generate rotationally symmetric and spiral optical beams. Our calculation method based on the Legendre transform parameterization of a caustic in terms of its tangent family provides a complementary and purely geometrical interpretation of the diffraction-integral method recently reported [21], and also shows naturally how there is in principle no paraxial limit to this technique when the phase profile is determined numerically. Using this technique, accelerating sheet caustic beams with parabolic, quartic and logarithmic profiles have been experimentally synthesized and experimental measurements have been shown to be in very good agreement with wave propagation calculations of an incident Gaussian beam subject to the calculated phase profile. Extension of our technique to higher dimension has yielded results displaying abrupt autofocussing behavior, and we have also shown compatibility with material processing applications using an imaging system to reduce the main intensity lobe at the caustic to sub-10 micron transverse dimension. In this regard, we note that, although our results on material processing have been performed in an intensity regime of only refractive index modification, scaling to higher intensities should readily lead to applications in material ablation and channel writing. We anticipate here that the arbitrary synthesis of accelerating caustic beams may provide a novel means to study and control how the intrinsic properties of nondiffracting beams are linked to the stabilization of optical propagation in regimes where nonlinear distortions of Gaussian beams are expected [27, 29].