## Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Recent constructions of nonparaxial accelerating beams cannot span more than a semicircle. Here, we present a ray based analysis for nonparaxial accelerating fields and pulses in two dimensions. We also develop a simple geometric procedure for finding mirror shapes that convert collimated fields or fields emanating from a point source into accelerating fields tracing circular caustics that extend well beyond a semicircle.

© 2014 Optical Society of America

## 1. Introduction

The term “accelerating beams” refers to beams that (at least approximately) preserve under propagation their transverse intensity profile, both in shape and scale, while their features follow curved trajectories [1]. Due to their properties, these beams have found applications ranging from particle and cell manipulation to laser micromachining [1]. The concept of optical accelerating beams was born within the paraxial regime with the one dimensional Airy beam [2] and the two-dimensional paraxial accelerating beams [3], following an analogy with similar solutions to the Schrödinger equation [4]. However, several recent publications [5–12] have been devoted to extending into the nonparaxial regime the concept of accelerating beams. By breaking away from the paraxial regime, these fields can trace large bending angles, and their transverse features can reach scales of the order of the wavelength.

Ray-optical treatments of accelerated field fields have been provided by several authors, e.g. [13, 15, 16], and the ray model has served as the basis for the design of fields whose intensity maxima trace arbitrary curves [6, 11, 14]. In this article and its follow-up [17], we give a ray-based description of nonparaxial accelerated fields, both monochromatic and pulsed, that approximately preserve their shape under propagation, as well as ways to generate them through mirrors. We start with the case of fields in two-dimensional space, where we provide a simple recipe for designing mirrors that transform collimated beams or fields generated by a point source into accelerated fields whose intensity maxima traces an arc that extends beyond a semicircle. These ideas are extended to three dimensions in [17], where caustics are not curves but surfaces, and where the intensity features trace circles or helicoids.

## 2. Accelerating beams as beams invariant under rotations

In this work we focus on the simpler situation of two-dimensional space with coordinates (*x*, *z*). In the paraxial regime, accelerating beams follow parabolic paths. This restriction in shape is easily understood in terms of the ray picture [4]. Accelerating beams such as Airy beams can be associated with families of rays that form caustics (ray envelopes), in the vicinity of which the main lobes of the intensity are located. The width of these intensity main lobes (as well as the slight displacement between their maximum and the caustic) depends both on the wavelength and the local curvature of the caustic [18].

In adopting the paraxial approximation in which *z* plays the role of the main propagation direction, one replaces the standard Euclidean geometry with a “paraxial geometry”. That is, the expression for the field must satisfy the paraxial wave equation
${\partial}_{x}^{2}U(x,z)+2\text{i}k{\partial}_{z}U(x,z)=0$ (where *k* is the wavenumber), and while this equation is invariant under translations in *x* and *z*, it is not invariant under rotations. It is instead invariant under “paraxial rotations” (i.e., translations in the transverse Fourier spectrum [19]) given by shears in *x* accompanied by phase factors: if *U*(*x*, *z*) is a paraxial solution, then so is *U*(*x* − *pz*, *z*)exp[i*k*(*px* − *p*^{2}*z*/2)], where *p* is the paraxial “angle” of rotation. That is, the paraxial rotation of a field with intensity *I*(*x*, *z*) = |*U*(*x*, *z*)|^{2} is a field with intensity *I*(*x* − *pz*, *z*). In this geometry, the curvature of a caustic (as a function of *z*) is given simply by its second derivative in *z* (which is invariant under paraxial rotations *x*, *z* → *x* − *pz*, *z*), so the only paths with constant curvature are parabolas whose axes are normal to *z*. This means that for the main intensity lobe to have constant width, the underlying caustic must be parabolic.

In the nonparaxial regime where the wave equation is invariant under the standard Euclidean transformations, on the other hand, circles are the only curves with constant curvature (defined in the standard Euclidean form [20]). Therefore, for an accelerated field to preserve the width of its main intensity lobe, this intensity lobe must follow a circular path. Notice, though, that in both the paraxial and nonparaxial regimes, choosing a shape with constant curvature for the caustic not only guarantees that the main intensity lobe maintains its transverse width, but also that the complete intensity structure is preserved over transverse lines (normal to *z* in the paraxial case, radial in the nonparaxial one). This is due to the fact that the complete ray distribution (which is fully determined by the caustic) is invariant under an appropriate combination of transformations. For the paraxial regime, such a transformation combination consists of a displacement and a paraxial rotation, while in the nonparaxial case it is a rotation around the circle’s center (see Fig. 1). As a result, the transverse distribution of all intensity maxima and minima remains also invariant, since the optical path difference between the two rays that cross at a given point outside the caustic depends only on the transverse distance from the point to the caustic.

## 3. Circular caustics and their generation through mirrors

The recent literature on nonparaxial accelerated fields or pulses tracing circular paths suggests that the main intensity maximum (and therefore the caustic) cannot span more than a semicircle due to limitations of forward propagation imposed by the use of optical systems composed of lenses and spatial light modulators. However, it is possible to convert light from a point source or a collimated beam onto a field with a caustic (circular or of any other convex shape) spanning an angle arbitrarily close to 2*π*. An easy way to achieve this is to use curved mirrors, which are capable of operating at much larger angles than lenses and spatial light modulators. Let us employ the simple construction depicted in Fig. 2(a), used often in the design of solar concentrators [21]. Cut an outline of the shape of the desired caustic in, say, thick cardboard, and place it on top of a flat piece of paper at the desired position. Then wind a piece of string around this cardboard shape, leaving plenty of extra string. Insert a pin at the desired point source location and tie the end of the string to it. [For an incident collimated field, the distant pin can be replaced by a rod with a sliding ring to which the string is tied, as in Fig. 2(b).] Now stretch the string with a pencil. Slide the pencil, keeping the string stretched, so that it unwinds from (or winds around) the cardboard shape. The resulting curve drawn on the paper is the shape of a mirror that reflects the light coming from a point source at the pin’s position into a caustic with the shape of the cardboard outline. Note that, for a given source position and caustic shape, there is a remaining degree of freedom corresponding to the length of the string. This degree of freedom controls the size of the mirror and affects the relative amplitude of the field at several parts of the caustic, as will be seen later.

Let us now focus on the case of a circular caustic of radius *R* (for which the reflected field’s intensity is approximately shape invariant), centered at the origin, and a collimated incident field traveling in the negative *z* direction (i.e., the source is at *z* = **∞**), as shown in Fig. 2(b). In this case, the string-based mental picture translates into the following simple parametric equation for the mirror over the (*x*, *z*) plane:

*ϕ*∈ (−

*π*,

*π*) is the angle between the

*z*axis and the reflected ray, and

*T*is the excess of string with respect to the distance of the source and the circle’s center, so that the point of lowest

*z*for the mirror equals −

*T*/2, and corresponds to

*x*= −

*R*,

*ϕ*= 0. In the limit

*T*≫

*R*, the mirror shape reduces to a parabola

*Z*=

*T*/2 − (

*X*+

*R*)

^{2}/2

*T*. Depending on how large the mirror is (namely, what range in

*ϕ*is used), the caustic can span a significant part of the circle (equal to the used range in

*ϕ*), coming arbitrarily close to closing it, even though the illumination is arriving from only one direction. Since this construction is completely geometrical and ignores the wavelength, there is no restriction of this mirror having to be small or large. Also, there is no “quantization” restriction on the ratio between the caustic perimeter and the wavelength, since the circle is not closed.

In order to achieve a fairly uniform amplitude over the circular caustic, the incident collimated field must be gently apodized to compensate for the angular variation in amplitude resulting from the nonlinear dependence of *X* in *ϕ*. Let *U*_{inc}(*x*) be the amplitude of the collimated illuminating field, and *A*(*ϕ*) be the angular spectrum of the reflected field. Then, to within a constant factor, conservation of power density requires |*U*_{in}(*X*)|^{2}d*X* = |*A*(*ϕ*)|^{2}d*ϕ*. (Notice that we are using the Debye approximation [22] in which a plane wave is assigned to each ray.) Requiring |*A*(*ϕ*)| to be constant over a range of directions implies that *U*_{inc}(*x*) (whose phase is constant) must be given according to

*T/R*(plotted parametrically with |

*U*

_{inc}|

^{2}and

*X*as functions of

*ϕ*). For large mirrors (

*T/R*≫ 1), (

*X*+

*R*)/

*T*≈ sin

*ϕ*/(1 + cos

*ϕ*) and $T{\left|{U}_{\text{inc}}\right|}^{2}\tilde{\propto}1+\text{cos}\varphi $, so the apodization’s shape tends to a decentered Lorentzian ${\left|{U}_{\text{inc}}\right|}^{2}\tilde{\propto}T/[{T}^{2}+{(x+R)}^{2}]$.

To model the wave field resulting from a single reflection off this mirror, we use a superposition of plane waves traveling in directions **u**(*ϕ*) = (sin*ϕ*, cos*ϕ*), weighted by the angular spectrum *A*(*ϕ*):

*A*(

*ϕ*) is given by

*kRϕ*, namely

*k*times the length of the string that winds/unwinds around the circular caustic in the string construction. As discussed earlier, the amplitude of the angular spectrum is related to the transverse profile of the incident field according to

*U*

_{inc}is chosen according to Eq. (2), then

*A*(

*ϕ*) ∝ exp(i

*kRϕ*), at least within a range in

*ϕ*.

Strictly speaking, one can argue that the presence of both the incident collimated field and of multiple reflections destroys the invariance along the caustic. However, the amplitude of the incident field can be significantly smaller than that of the circular caustic, especially for large *T/R*. This can be appreciated from Fig. 4(a), where the incident rays are shown in yellow and the rays following a first reflection are shown in green. The amplitude of each component is proportional to the square root of the density of rays. If we fix the angular density of the green rays, the density of yellow rays decreases for increasing *T/R*, meaning that the relative importance of the incident field decreases. Also shown are the rays following a second reflection (orange) which are seen to stay clear of the caustic. Note that not all rays hit the finite section of the mirror being used a second time, and those that do are separated into two bundles (both shown in orange) after the second reflection: (i) The bundle on the left correspond to rays that were incident on the right-hand side of the mirror and touched the caustic on its lower half after the first reflection. It is easy to see that neither these rays nor their subsequent reflections can ever return to the caustic. (ii) The bundle on the right result from rays that were incident on the left-hand side of the mirror and touched the caustic on its upper half. These rays also stay away from the caustic provided the mirror accepts only incident rays with *x* ≥ *X*(*ϕ*_{c}), where *ϕ _{c}* is the solution of

*ϕ*corresponds to the angle after the first reflection of the ray that goes on to hit the mirror normally on its second reflection, and that therefore retroreflects and touches the caustic a second time. This ray is shown in blue in Figs. 4(b,c) for the case where a significantly larger segment of the mirror is used. Rays incident on the mirror at

_{c}*x*>

*X*(

*ϕ*

_{c}) do not touch the caustic following the second (or any subsequent) reflection. On the other hand, rays incident on the mirror at

*x*<

*X*(

*ϕ*

_{c}) do cross the circular caustic after their second reflection, but they do not form a second caustic that overlaps with the first, and their spacing (and hence their disruption to the caustic pattern) is comparable to that of the incident rays. For the case shown in Fig. 4,

*X*(

*ϕ*

_{c}) = −27.64

*R*, so one would have to use a mirror segment significantly larger than the caustic for multiple-reflected rays to overlap with the caustic. As will be discussed in [17], one can design mirrors in three dimensions for which both the incident and multiply reflected rays stay away from the caustic.

Figure 5 shows (a) the estimate of the intensity of the reflected field in the vicinity of the caustic (calculated through a uniform approximation given below) for a mirror collecting light where |*A*(*ϕ*)| = exp[−(1 − cos*ϕ*)^{10}/1.7^{10}] (i.e., the angular spectrum’s magnitude is nearly constant for |*ϕ*| < 0.6*π* and then tapers down to zero), and (b) the intensity for the same field when we include the incident field before reflection in the case where *T* = 6.2*R*. For larger *T*, the effects caused by the presence of the incident field become less visible, as discussed earlier. The effects of multiple reflections are negligible in the region shown here.

## 4. Pulsed solutions

The method described above works well not only for monochromatic fields but also for pulsed ones, i.e., it generates “light bullets” that can in principle trace nearly a full circle. Further, in the case of pulses, the incident field can be made to not coincide temporally with the reflected bullet. The movie associated with Fig. 2 illustrates how this works. The green circles represent the pulse’s maximum along each ray. Note that different rays give rise to the maximum at the caustic at different times. If we assume that the medium and mirror are not dispersive, the field can be modeled as a pulsed version of the Debye superposition (normally applied to monochromatic fields [22]), in which a suitably retarded pulsed plane wave is associated with each ray:

*P*is an arbitrary pulse form and

*c*is the speed of light. Let us use a (normalized) Gaussian pulse of spatial width

*w*: where

*k*

_{0}is the pulse’s central wavenumber. The reflected pulse can be estimated by using an asymptotic technique known as a uniform approximation [23], in which the integral expression for the field is replaced by an expression involving Airy functions and their derivatives, whose integral expression has the same saddle points as those of the field. The resulting expression is

*r*,

*φ*) are the polar coordinates of

**r**, Ai is the Airy function of the first kind, Ai′ is its derivative, and

*A*(

*ϕ*)| does not vary significantly over angular variations of magnitude

*k*

_{0}

*R*. Note also that the monochromatic case, shown in Fig. 5, corresponds to the limit of large spatial width

*w*. For this case, the contribution proportional to Ai is the leading one along regions of the caustic where the amplitude is fairly constant, while the contribution proportional to Ai′ provides corrections near the ends of the caustic corresponding to regions of space where the parts of the field approaching the caustic and departing from it are of significantly unequal magnitudes. For pulsed beams (short spatial width

*w*) both contributions to the field estimate are important.

Figure 6 and Media 5 show a pulse running along a large segment of the circular caustic. Note that instead of an isolated pulse, one could send in a collimated pulse train whose pulse separation equals the circular caustic’s perimeter. For a sufficiently large mirror, this will create the illusion of a pulse continuously running around the circle (except for a small interruption).

## 5. Concluding remarks

Given the focus of this work on fields that preserve their transverse intensity profile, we concentrated on the case of circular caustics. However, it is worth commenting briefly on other caustic shapes that can be generated by using this construction. The main restriction in the shape of a caustic produced by a curved mirror is that it cannot present inflection points, as mathematically these would cause the resulting mirror to have cusps and block itself. It is then tempting to state that the caustic must have a convex shape. However, the caustic can also be non-convex if it is composed of two convex segments joined at a cusp, as is the case of the well-known cardioid caustic formed by a circular mirror, visible on the surface of a milky drink in a cylindrical mug when the sun is shining on it. (In this case the end of the string should be attached to the cusp, so that the string wraps around one or the other caustic segment.) Some convex caustic shapes of interest include ellipses or parabolas, as these are associated, respectively, with Mathieu [7, 9] and Weber [7, 8] fields (given the appropriate apodization of the field incident on the mirror), expressible analytically through separation of variables. Figure 7 shows mirrors that generate elliptical caustics with different orientations, both tracing an angle beyond *π*. The density of the incident rays reflects the intensity apodization needed to achieve a field resembling a Mathieu field.

In the second part of this series [17], we will extend these ideas to three dimensional accelerating fields whose caustic sheets can describe a series of shapes, and whose intensity features trace either circles or helicoids.

## Acknowledgments

MAA acknowledges support from the National Science Foundation ( PHY-1068325).

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