We present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey beam displays properties similar not only to Airy beams but also Gaussian and Bessel beams. These properties include an inherent auto-focusing effect, as well as form-invariance on propagation and self-healing. We describe the theory of propagating Pearcey beams and present experimental verification of their auto-focusing and self-healing behaviour.
© 2012 Optical Society of America
Special functions have in recent years enjoyed a vogue in the classical optics of paraxial beam propagation, owing to their comparative ease of generation, straightforward mathematical representation and, in many cases, structural stability on propagation. Three important examples are the Gaussian beams, , Bessel beams [2,3] and Airy beams [4,5]. Many generalizations of these beams have been studied, including, for instance, Hermite-Gaussian, Laguerre-Gaussian and Ince-Gaussian beams [1, 6], Mathieu beams , and accelerating parabolic beams , all of which exhibit properties characteristic of their respective beam families. Here, we describe a new kind of paraxial beam – the Pearcey beam – based on the Pearcey function of catastrophe theory [9,10], which describes diffraction about a cusp caustic and occurs as a two-dimensional counterpart of the Airy function within the catastrophe theory framework . The propagating form of the Pearcey beam has several noteworthy properties, some of which are reminiscent not only of Airy beams, but Gaussian and Bessel beams as well. Most remarkably though, the Pearcey beam possesses an additional auto-focusing property on propagation.
The properties of Gaussian beams, as propagating laser cavity modes, are extremely familiar, so some of their striking properties may seem unremarkable. The intensity pattern of a Gaussian beam is invariant upon propagation, apart from an overall transverse scaling. We refer to such beams as form-invariant, where the intensity pattern is unchanged up to a similarity transformation (i.e. z-dependent translation, rotation or scaling). Furthermore, starting from the focal plane, a Gaussian beam spreads to its Rayleigh distance zR, after which the propagation of the field is effectively indistinguishable from that originating from a point source. Mathematically, the Gaussian beam can then be written as the light from this point source translated in the imaginary z-direction [1,12]. The product of the width (second moment) of the Gaussian in the focal plane and in the Fourier plane is the smallest possible for any beam, from the uncertainty principle. Thus, a wider range of Fourier components – a greater spectral width – leads to tighter focusing.
Bessel beams are not only form-invariant, but completely invariant upon propagation (apart from an overall phase factor), as a result of their Fourier transform being concentrated on a δ-function ring with fixed wavenumber kr. For this reason, Bessel beams are sometimes described as nondiffracting [2,13] and it is natural to study how the presence of opaque obstacles affect the beam’s properties. Indeed, Bessel beams display self-healing over relatively small propagation distances [14–16], making them useful for optical trapping and applications through weakly dispersive media . Laguerre-Gaussian beams also display self-healing, but at much longer distances, comparable to the Rayleigh distance . Since Bessel functions are not strictly normalizable (as their Fourier transforms are δ-concentrated), physical realizations require some modification of a pure Bessel function, achieved by modulating the Bessel function amplitude with a wide Gaussian function in real space, which incurs some diffractive spreading .
The profiles of Airy beams, similar to Bessel beams, do not change at all on propagation, although the Airy amplitude pattern accelerates in the transverse direction under free-space propagation. These remarkable features have, in recent years, led to intense investigation of Airy beams [4, 5, 20] and their self-healing properties [21, 22]. Physical realizations similarly require modulation of the Airy function, whose total intensity is infinite in the transverse direction.
Historically, Airy functions were introduced by George Biddell Airy  to describe the interference fringes sometimes observable below a rainbow, i.e. ‘supernumerary rainbows’ . They occur more generally as the characteristic diffraction pattern around a caustic of codimension 1 (i.e. a ‘fold’, which is a line in two dimensions), a fact which has been used to generate beams of cubic and even exponential curvature . Codimension 2 caustics are generically cusps , and the characteristic diffraction pattern around them is the so-called Pearcey function [24,26], described below. More complicated caustics occur in three and higher dimensions, classified and understood mathematically via catastrophe theory  including their topologies and characteristic diffraction patterns [11,24,27].
The properties of paraxially propagating Airy beams do not seem directly related to their connection with diffraction catastrophes, beyond the observation that caustics, like propagating Airy beams, are curved . It is therefore natural to ask whether the next member of the catastrophe optics hierarchy, the Pearcey function, has unusual propagation properties. We investigate such a beam here, and show, in addition to its self-healing and form-invariance upon propagation, the Pearcey beam auto-focuses in a remarkable way at a propagation distance mathematically analogous to the Rayleigh distance of a Gaussian beam, and the pattern spatially inverts beyond this focus. Like the Bessel beam and Airy beam, a pure Pearcey beam has infinite energy, but can be made finite by modulating the Pearcey function with a Gaussian in real space, which does not significantly change the beam’s properties. We formulate the Pearcey beam analytically and present experimental results of the finite energy Pearcey-Gauss beam generation using computer-controlled holograms, corroborating the properties we describe.
In catastrophe optics, the Pearcey function represents the diffraction about a focus in 1 + 1 dimensions, which experiences spherical aberration – the rays underlying the diffraction caustic form a cusp either below or above the geometrical focus depending on the sign of the spherical aberration. In a sense, the propagating Pearcey beam simulates such a 1 + 1 dimensional aberrated focus in each transverse plane, with different amounts of aberration (positive or negative) for different z. This picture accounts for the form-invariance of the Pearcey beam, since each different amount of aberration gives a differently-scaled Pearcey function, and the auto-focusing plane is the plane in which the aberration is zero, i.e. it looks like the 1+1 propagation pattern of a focused beam. The beam inversion occurs beyond this distance when the aberration has changed sign.
The structure of this paper is as follows. In section 2 we mathematically construct the analytic form of the paraxially propagating Pearcey beam and the more physical Pearcey-Gauss beam, describing their auto-focusing and associated inversion. In section 3 we investigate the nature of the focal spot formed by the finite energy Pearcey-Gauss beam and in section 4 we present experimental results, demonstrating the beam’s auto-focusing and self-healing properties. We conclude in section 5 with further discussion and outlook.
2. The Pearcey beamFig. 1(a) for x0 = y0 = 0.1 mm, along with the cusp caustic which underlies the amplitude pattern. Equation (1) can be calculated numerically using a contour rotation in the complex s plane, s → s′eiπ/8, which guarantees convergence of the integral as s′ → ±∞. The mathematical definition of the Airy function is similar; the quartic polynomial exponent in Eq. (1) is replaced by a cubic s3 + sx/x0, which has a fold singularity at the origin, separating a two-wave interference pattern from an exponentially decaying region with no waves .
The form of the cusp which underpins the intensity pattern can be expressed  as the (x, y) points satisfyingFig. 1(a). The values of x0 and y0 determine the cartesian scalings of the cusp, which itself is important in the anatomy of the Pearcey function; the line demarcates two interference regimes in the pattern, corresponding to a single wave and three wave interference. The Pearcey function of Eq. (1) can be understood mathematically by the contributing saddle points from the integral in the complex s-plane. The cusp is then the locus of coalescence of these saddle points, but in this case the saddles do not correspond directly to optical rays, which is different from the occurrence of a cusp in geometric catastrophe optics [11,24].
The Fourier transform of the Pearcey function can be expressed asEquation (3) thus describes a δ-line concentrated on the parabola , modulated by a kx-dependent phase factor. Despite having real-space representations closely related by catastrophe optics, the Fourier transforms of the Pearcey and Airy functions differ - the latter has a uniform amplitude and cubic phase dependence. The concentration on a curve in Fourier space is more reminiscent of the Fourier representation of a Bessel beam, although the magnitude of the Pearcey wave vectors is not fixed (and hence the beam profile changes on propagation).
An analytic form of the propagating monochromatic Pearcey beam can be found using the paraxial, scalar Huygens-Fresnel integral  for an initial field (at z = 0) given by the Pearcey function Eq. (1). This takes the formEquation (4) and Eq. (2) may be combined to give an expression for the evolution of the cusp in z,
Figures 1(a)–1(f) show the intensity pattern of the Pearcey beam as z increases. The rate of scaling becomes infinitely fast as z approaches ze but changes more quickly in the y-direction than in x. As a result, the bright fold catastrophe ‘arms’ of the Pearcey pattern appear to lift upwards towards the line , and the intensity becomes more concentrated as z → ze. We see from Eq. (4) that this occurs because there is a singularity in both x and y at z = ze, so the scaling is effectively infinite (which is possible since the total Pearcey pattern cannot be normalized). When z > ze, Eq. (5) has a real solution with an inverted cusp mirroring the behavior of the cusp when z < ze, and the Pearcey function follows the inversion of the cusp, as evident from Fig. 1; this spatial inversion is of course expected for a light beam passing through a focus. In the ‘focusing’ plane the beam is singular, taking the form
The singularity can be understood mathematically using a complex analysis approach. When x0 and y0 are replaced with the z-dependent scalings from Eq. (4), the Pearcey integrand of Eq. (1) can be considered as a complex function of z. This function has an essential singularity at z = ze, i.e. a reciprocal-type singularity in the exponent, which is real-valued, accompanied by a branch point singularity multiplying the exponential. This contrasts with the propagating form of a Gaussian beam
In the Gaussian beam, there is an essential singularity at izR, interpreted physically as a ‘complex source’ [1, 12]. In this interpretation, the Gaussian waist corresponds to the best possible image of this complex point source, achieved when it is closest to the real z-axis. The Pearcey beam integrand, on the other hand, has its singularity on the real line, and effectively passes through it upon propagation. The form of the singularity is more complicated in the Pearcey beam case, and the corresponding pattern in the singular plane z = ze is completely delocalized in x. Therefore, the limits imposed on the transverse scalings in the Gaussian case do not apply for the Pearcey beam and we see from Eq. (4) that as z → ze the factors scaling x and y tend to infinity, highly concentrating the intensity of the beam to a line in this particular z-plane. We refer to this phenomenon as essential focusing – a high concentration of intensity due to the existence of a real-valued essential singularity in the z-dependence. In a sense, this is analogous to limiting the waist width and hence the Rayleigh distance of a Gaussian beam to zero, making the Gaussian’s ‘complex source’ real-valued and the beam infinitely localized in the waist plane. Therefore, the bright line from the real essential singularity (described by Eq. (6)) is, in a sense, a ‘source’ for the propagating Pearcey pattern.
Of course, the Pearcey beam’s essential singularity is not physical. For the paraxial approximation to be valid it is necessary that the transverse spatial spectrum is small so that |kx|, |ky| ≪ k, but from Eq. (3) it is evident that such limitations do not exist for the Pearcey beam, giving rise to the observed singular behaviour. Fortunately, as with the Airy beam, the rapidly oscillating phases at large kx and ky do not contribute due to cancellation, and it is possible to experimentally realise an approximation of the Pearcey beam from the spectrum near the origin. A natural regularization is to modulate the initial Pearcey pattern by a Gaussian with large width w0, which gives the Pearcey pattern finite total intensity whilst maintaining its analytic tractability and limiting its spectrum. In Eq. (4), it is easy to include a Gaussian in the intensity profile and integrate as before (or alternatively via the ABCD approach of Ref. ). Therefore the propagating form of this finite energy Pearcey-Gauss beam isEq. (7). As with many analytic forms of beams made from modulating a function by a Gaussian, the Pearcey-Gauss beam has a complex argument upon propagation in z, giving rise to the complex analytic continuation of the Pearcey function, which maintains a form very similar to a Pearcey function with real arguments.
Figure 2 shows the Pearcey-Gauss intensity pattern from Eq. (8). Evidently, the Gaussian modulation does not affect the main behavior of the Pearcey beam – the same arm-straightening, auto-focusing and inversion effects are evident as for the Pearcey beam of Eq. (4). However, in the modulated case, the additional imaginary part of ζ(z) in Eq. (8) gives the problematic essential singularity a very small imaginary part. The increase of the x and y scalings still occurs, as does the inversion, but now with finite intensity across the beam and a slight deformation to the original Pearcey function with real arguments. In the new essential focusing plane, the infinite line becomes a small spot centred in an hourglass-shaped profile as depicted in Fig. 2(d). That is, modulation with a Gaussian has made the essential singularity ‘source’ complex, but in this case it is closer than one Rayleigh distance to the real z-axis. The result is a small anisotropic spot of high intensity, shown in Fig. 2(d).
The Fourier transform of the Pearcey-Gauss beam can be calculated by convolving Eq. (3) with the Fourier transform of the modulating Gaussian, according to the convolution theorem. This gives the expressionEq. (3), despite the fact it can be expressed as a Pearcey function. Figure 3(b) shows the intensity of Eq. (9) for the parameters of Fig. 2 (x0 = y0 = 0.1 mm, w0 = 2.0 mm and λ = 500 nm).
3. The Pearcey-Gauss focus
In the introduction, we proposed the interpretation that each two-dimensional transverse plane of the Pearcey beam can be imagined as a ‘simulation’ of a spherically aberrated one-dimensional wavepacket in x, being focused as it ‘propagates’ in y. That is, the transverse plane of the beam appears to be a longitudinal section of a spherically aberrated focus. The amount of aberration depends on z, and in the essential focusing plane (z = ze) the beam simulates the pattern around a point focus with zero aberration. This explains the ‘hourglass’ shape of the intensity distribution in Fig. 2(d). As w0 (the width of the modulating Gaussian in Eq. (8)) increases, the Pearcey-Gauss beam becomes more like the infinite-energy Pearcey beam and – in the limit of infinite w0 – the essential focusing plane becomes a section of a focus of unit numerical aperture (assuming a refractive index of 1). However, before this limit is reached, the essential focus has an intensity peak with a narrow FWHM in both the x- and y-directions.
Closed form expressions for the shape of the essential focus along the x- and y-directions are in fact possible and respectively take the form30].
Figure 3(a) shows a magnification of the essential focus of the Pearcey-Gauss beam – Eq. (8) for z = ze. Figures 3(c) and 3(d) show the intensity cross-sections of the essential focus given by Eqs. (10) and (11) respectively. The width of the essential focus in the x- and y-directions is not significantly smaller than that of an astigmatic Gaussian of equal spectral width.
We note here that the dark regions in the neck of the hourglass display significant super-oscillatory behaviour , in keeping with the understanding that narrow focal spots are often bounded by highly super-oscillatory regions [32–34].
4. Experimental realization of the Pearcey beam
Our experimental realization of a Pearcey beam (or, more accurately, a Pearcey-Gauss beam) involved encoding a Pearcey-Gauss spectrum (as in Eq. (9)) onto a spatial light modulator (SLM). The image encoded onto the SLM is shown in Fig. 4(b), and was incorporated into the experimental set up as described in Fig. 4(a). A helium-neon laser source (λ = 633nm, Pmax = 5mW) was used and the beam was sent through a 50μm pinhole in order to obtain an initially homogeneous Gaussian intensity profile. The beam was then expanded with a telescope (L1 and L2) in order to slightly overfill the SLM (a Holoeye LC-R 2500). The SLM operated in the standard first-order diffraction configuration and the Pearcey beam was created in the far-field of the SLM. In order to filter the first order beam carrying the Pearcey pattern from the unmodulated zero-order beam, a pinhole aperture was located in the back focal plane of lens L3. To study the consecutive transverse cross-sections of the resulting Pearcey beam, another telescope (lenses L4 and L5) was used to image it on the obstacle and subsequently on a CCD camera (Basler piA640-210gm, pixel size: 7.4μm × 7.4μm), in order to record the images. The CCD camera was in the Fourier plane of the SLM and was mounted on a translation stage in order to get images over the propagation range of the beam.
Figure 5 shows the experimentally obtained images for the propagation distances indicated. These results support the theoretical predictions, exhibiting inversion as well as auto-focusing at the essential focus plane. We note that the precise position of the focus is dependent on the experimental setup. Figure 5(d) shows the essential focus, which is similar to that shown in Fig. 2(d).
We also demonstrated experimentally the capability of the Pearcey beam to self-heal. Figure 6 shows the experimental results of applying a rectangular obstacle (indicated by the white line in Fig. 6(a)) to the z = 0 plane of the Pearcey-Gauss beam. These results clearly display a revival of the Pearcey pattern after the essential focusing, which is still strongly evident in the propagation of the beam. Numerical results for small perturbations to the Pearcey-Gauss beam display pattern revival after a distance of as little as 5 cm. Larger perturbations cause a greater disturbance and recovery of the beam profile requires propagation over a longer distance; for example, blocking the main intensity lobe of the beam revives after a propagation distance of the order of 20 cm. These lengths are similar to those of an Airy beam for comparable parameters and obstructions . Significantly, the Pearcey-Gauss beam’s profile recovers close to its original form after the inversion for all but the most severe disturbances. We conclude that, similar to Airy and Bessel beams, small perturbations to the field of the Pearcey beam do not drastically effect the intensity pattern at greater propagation distances, especially not after the inversion plane.
It is not clear how to define the concept of self-healing rigorously, although it is clear conceptually. A detailed study of self-healing is outside the scope of the present paper, but we note that since the Pearcey beam is not nondiffracting and is self-healing, there is compelling reason to decouple these two concepts – self-healing beams are not, by necessity, nondiffracting (although the opposite may be typical) [18, 35]. This is particularly striking in the case of the Pearcey beam, as its cross-section consists of a complicated interference pattern, which is robust – apart from rescaling – both to propagation and disruption by small obstacles.
We have introduced a new beam profile based on the Pearcey function of catastrophe optics. Its transverse profile looks like a diffraction cusp, which, upon propagation, auto-focuses to a small spot centred in an hourglass-shaped intensity distribution. The beam profile then spatially inverts after this plane.
Our analysis has highlighted similarities of this propagation to three well-known beams of paraxial optics: Gaussian, Bessel and Airy beams. The definition of the Pearcey function and its relation to caustics is similar to the Airy function, and, like a Gaussian, it is form-invariant with a focus given by an essential singularity (which is real for a Pearcey beam, and imaginary for a Gaussian). Also, the Pearcey beam’s spatial spectrum is concentrated on a δ-parabola, similar to the δ-ring of a Bessel beam’s Fourier transform. Simple superpositions of Pearcey beams would therefore be easily possible because of the small bright area required on an SLM to form them. The Pearcey beam also exhibits self-healing properties analogously to Airy and Bessel beams, making it robust to local obstructions and potentially useful for cell or micro-particle manipulation but notably is not propagation invariant (non-diffracting).
The essential focusing exhibited by the Pearcey beam is a generalization of the complex source approach to Gaussian beams, with an integral representation whose integrand has a real-valued ‘source’, giving rise to a singular focusing plane. We note that although essential focusing is superficially similar to abrupt autofocusing [36,37], they are mathematically distinct. We also note that there is some similarity to the so-called ‘caustic beam’ , which appears to be based on an astroid of four cusps, but which does not display any auto-focusing.
Mathematically, the essential focusing we have described for Pearcey beams will occur for any transverse amplitude distribution given by a function of the form
We are grateful for discussions with John Hannay, Jörg Götte, Anna Khoroshun and Stefan Skupin. We gratefully acknowledge funding from the EPSRC. KD is a Royal Society Wolfson Merit Award holder and MRD is a Royal Society University Research Fellow.
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