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1 ACMAC s PrePrint Repository Abruptly autofousing and autodefousing optial beams with arbitrary austis Ioannis Chremmos and Zhigang Chen and Demetrios N. Christodoulides and Nikolaos K. Efremidis Original Citation: Chremmos, Ioannis and Chen, Zhigang and Christodoulides, Demetrios N. and Efremidis, Nikolaos K. (1) Abruptly autofousing and autodefousing optial beams with arbitrary austis. Physial Review A. ISSN (In Press) This version is available at: Available in ACMAC s PrePrint Repository: February 1 ACMAC s PrePrint Repository aim is to enable open aess to the sholarly output of ACMAC.

2 Abruptly autofousing and autodefousing optial beams with arbitrary austis Ioannis D. Chremmos 1, Zhigang Chen, Demetrios N. Christodoulides 3, and Nikolaos K. Efremidis 1 1 Department of Applied Mathematis, University of Crete, Heraklion 7149, Crete, Greee Department of Physis and Astronomy, San Franiso State University, San Franiso, CA CREOL/College of Optis, University of Central Florida, Orlando, Florida 3816 Corresponding author: johremm@entral.ntua.gr Abstrat: We propose a simple yet effiient method for generating abruptly autofousing optial beams with arbitrary austis. In addition we introdue a family of abruptly autodefousing beams whose maximum intensity suddenly dereases by orders of magnitude right after the target. The method relies on appropriately modulating the phase of a irularly symmetri optial wavefront, suh as that of a Gaussian, and subsequently on Fourier-transforming it by means of a lens. If two suh beams are superimposed in a Bessel-like standing wave pattern, then a omplete mirror-symmetri, with respet to the foal plane, austi surfae of revolution is formed that an be used as an optial bottle. We also show how the same method an be used to produe aelerating 1D or D optial beams with arbitrary onvex austis.

3 1. Introdution Reently, a family of optial waves was introdued with abruptly autofousing (AAF) properties [1]. The new waves (also termed irular Airy beams - CAB) have a irularly symmetri initial amplitude that osillates outward of a dark disk like an exponentially trunated Airy funtion. By virtue of the two outstanding features of finite-power Airy beams [,3], namely their self aeleration and their resistane to diffration, this eventually results in light beams, that an propagate over several Rayleigh lengths with minimum shape distortion and almost onstant maximum intensity, until an abrupt fousing takes plae right before a target, where the intensity is suddenly enhaned by orders of magnitude. These theoretial preditions were subsequently verified by experimental observations [4,5]. These observations also demonstrated that AAF beams ould outperform standard Gaussian beams, espeially in settings where high intensity ontrasts must be delivered under onditions involving long foal-distane-to-aperture ratios (f-numbers) [4]. In addition, this silent or low intensity mode, at whih AAF waves approah their target, makes them ideal andidates for medial laser appliations where ollateral tissue damage is supposed to be kept at a minimum. Other possible appliations inlude laser waveguide writing in bulk glasses and partile trapping and guiding [5]. The fousing mehanism of AAF beams is fundamentally different from that of Gaussian beams. In the latter ase, the wave s onstituent rays form a sharpened penil that onverges at a single point, the fous. As the beam s ross-setional area gradually dereases, the maximum intensity over the transverse plane inreases in a Lorentzian fashion, entered at the fous. In the ase of AAF beams however, the rays responsible for fousing are emitted from the exterior of a dark disk on the input plane and stay tangent to a onvex austi surfae of revolution (SoR) that ontrats toward the beam axis [1]. By virtue of its Airy transverse amplitude profile, this austi is intrinsially diffration-resisting, therefore keeping its maximum intensity almost onstant during propagation and its interior almost void of optial energy. Fousing ours as a result of an on-axis ollapse of this SoR and is so abrupt as the transition from the dark to the lit side of an optial austi. At the point of ollapse, the rays emitted from a ertain irle on the input plane interfere onstrutively and the gradient of the field amplitude is maximized.

4 Building on the onept of a ollapsing austi SoR, the AAF wave family was reently broadened to inlude general power-law austis that evolve from a sublinearly hirped input amplitude [6]. In this latter work, we showed that a ν th -power austi requires the input amplitude to osillate with a hirp of the order 1, whih generalizes the ase of CABs, whose paraboli trajetory is a result of the 3 / hirp of the Airy funtion itself. Although not non-diffrating as ideal CABs, these pre-engineered beams were shown to exhibit attrative features, suh as enhaned fousing abruptness, larger intensity ontrasts and suppressed post-foal intensity maxima. To benefit from the attrative properties of AAF waves, it is ruial to generate them effiiently. This is generally a nontrivial task, sine these waves evolve from initial amplitudes that are not easy to implement diretly, suh as onentri Airy rings. A possible alternative is to generate the Fourier transform (FT) of the initial ondition first and then inverse-fourier transform it by means of a lens. A similar approah was adopted in the first demonstration of CABs [4], where the initial wavefront was produed by enoding both amplitude and phase information on a phase-only filter. The FT tehnique was also employed in [7], where a hologram of the FT was produed. In addition in this latter work, the FT of CABs was treated analytially and found to behave like a Bessel funtion whose argument is enhaned by a ubi phase term, i.e. a quadrati hirp. By tuning the strength of the hirp relative to the lens foal distane, it was possible to generate AAF beams with two foi, the one being defousing while the other one fousing, thereby defining the ends of an elegant paraboloid optial bottle [7]. From the above it is lear that produing AAF beams an be equally diffiult, either in the real or in the Fourier spae, sine, in both ases, a ompliated initial ondition, varying both in phase and in amplitude, must be produed. However, when the requirement for the exat generation of a speifi AAF wave an be relaxed, the implementation proedure an be signifiantly simplified. Indeed, having explained the AAF mehanism through both ray and wave optis [1,6], one realizes that the ritial phenomenon is the formation of the austi and depends primarily on the phase modulation of the input wavefront and seondarily on the envelope of its amplitude. This fat has already failitated the generation of arbitrary onvex 1D or D beams, by applying the appropriate 1D or D phase mask on a plane wave [9] or

5 on a Gaussian beam [1]. This is perhaps the most obvious way to diretly produe aelerating 1D or D austis. However, in some appliations, it ould be more advantageous to apply the same onept in the Fourier spae. A major reason is that the FT lens provides an additional degree of freedom for easily targeting and sizing the generated beam [7]. Indeed, as noted in the latter paper, the optial bottles produed with the FT method an be arbitrarily saled without losing their symmetry or having to modify the input ondition, by simply hanging the lens foal distane. This is a true advantage of the FT approah ompared to the diret (real-spae) approah, whih requires that all beam harateristis must be presribed on the phase mask, while even a simple saling of the beam requires to redesign the phase modulation. In addition, the lens offers a physial f separation between objet and image planes, whih ould be useful in appliations when the phase mask annot be positioned very lose to the target. In this work, we present a simple yet general method for generating in the Fourier spae AAF or abruptly autodefousing (AADF) optial beams with preengineered austi SoR. From a ertain viewpoint, this generalizes our previous work on CABs and their paraboli austis [7] to AAF waves with arbitrary onvex austis. We hereby show that suh beams an be produed effiiently by simple means of a irularly symmetri phase mask and a FT lens, ating suessively on an optial wavefront with no partiular amplitude information. The required phase is the sum of a linear and a nonlinear term. The linear term is responsible for reating an annular fousing pattern on the image plane, while the nonlinear term is responsible for transforming this pattern into an Airy pattern thus determining the shape of the austi produed before or after that plane. Speial attention is paid to the ase of power-law austis, for whih analytial results are readily obtained and reported. In partiular, we find that a ν th -power austi requires a n th -power phase with the orders being related as n 1 / 1. Counter-intuitively, for a given ν, the stronger the nonlinear phase the weaker is the aeleration of the austi, hene the longer is the distane from the image plane to the target. The linear phase and the lens foal plane determine the size of the produed AAF beam in terms of its width at the image plane, and also in terms of the distane from that plane to the target. In Setion, we desribe the proposed method analytially and subsequently, in Setion 3, we support it with numerial alulations that demonstrate the

6 generation of AAF and AADF beams and also that of optial bottles with sinusoidal shape. Although our fous is on AAF waves, we omplete this paper by showing that the same method an atually be used to produe aelerating 1D or D beams with arbitrary onvex trajetories.. Engineering AAF/AADF waves in the Fourier spae To begin, onsider the amplitude of a phase-modulated irularly symmetri wavefront u r A r exp i r A r exp iar iq r, (1) where r is the polar distane normalized by an arbitrary length, say x, envelope and Ar is a real r is the phase onsisting of a linear term with slope a and a nonlinear (e.g. power-law) term qr. Suh an initial ondition an be easily realized by refleting a plane wave or a ollimated Gaussian beam on the fae of a spatial light modulator (SLM) programmed with a phase r. Subsequently, let us examine the evolution of this wave through the single-lens FT system of Fig. 1 under the validity of the paraxial approximation z t u i u, where subsript t stands for transverse and z is the propagation distane normalized by x /, λ being the optial wavelength. Propagating the waves before and after the lens aording to the Fresnel diffration integral [6], and taking into aount the quadrati phase exp( ir / f ) imprinted on the wave transmitted through the lens, it an be shown that the optial field on the image plane ( z f, where f is the foal length) reads where u r, z f i / f U r / f, () U k u J k d (3) is the Hankel transform of the objet wavefuntion u. Obviously, Eq. () expresses the FT property of the lens. Substituting Eq. (3) into Eq. () and using the familiar integral representation of Bessel funtion

7 1 J x expix os d, (4) we obtain i r f f ur, z f exp ia iq i os dd, (5) where u was substituted from Eq. (1). Even in its simplified form of Eq. (3), this integral annot be evaluated analytially for a general funtion q. Hene it is reasonable to resort to a stationary-phase (SP) omputation, whih is justified by the osillatory nature of the integrand. Assuming that q for all, where the prime denotes the derivative with respet to the argument, it is readily seen that there is only one stationary point s, s r,, where r is the solution of equation a q r r / f. (6) After some algebra, the result of integration in the neighbourhood of, is s s SP r rr u r, z f r exp iar iqr i. fq r r f (7) The last two equations lead to two important onlusions: First, Eq. (6) implies r af, whih means in essene that only points on the image plane lying outside that disk plane have appreiable amplitude. Seondly, sine r is a funtion of the observation point r, Eq. (7) shows that the wave amplitude on the image plane is also nonlinearly phase-modulated. Differentiating the phase of Eq. (7) with respet to r and using Eq. (6), one obtains r / f, i.e. the phase modulation is of onverging nature. Therefore, the wave on the image plane satisfies the preonditions for evolving into an AAF wave. If the phase of Eq. (7) is properly designed, then an inward bending austi SoR with initial width af will be formed and eventually fous abruptly somewhere in the half-spae z f. Further understanding of this proess an be gained through a ray optis interpretation of the propagation dynamis. Referring to Fig. 1, let us follow the path of the ray starting from point r, on the input plane. Aording to Eq. (1), this ray

8 travels at an angle with the z-axis and reahes the surfae of the lens (whih is assumed to be infinitesimally thin) at r 1, f, s where r 1 r fs and tan r (8) is the orresponding slope. Passing through the lens, the ray deflets inwards and emerges from point r 1, f, with a modified slope s1 tan 1 s r1 / f. The transmitted ray rosses the image plane at r r1 fs1 with a slope tan s s1. Combining the above we obtain the equations onneting the exit position and slope of a ray r fs, s r / f, (9) r,s. r,s to the input values Equations (9) are equivalent to the onlusions reahed previously, noting that the point here named r is the observation point r of Eqs. (6) and (7). As r,s r vary ontinuously along the input plane, the transmitted rays form a austi that is expressed with oordinates r,, where r is the radial distane, z f is the distane from the image plane and r, is interpreted as the point at whih ray r,s touhes the austi. From Fig. 1 obvious are also the equations s r, r r s. (1) Differentiating the seond of Eqs. (1) with respet to and using Eqs. (9), it an be shown that the austi is expressed in terms of the input ray harateristis as where r fs r fr s r f s r,,, (11) r serves as a parameter and s r is given by Eq. (8). Equations (8) and (11) provide the means for a diret design approah, namely to determine the austi resulting from a given input phase modulation. Alternatively, one ould work inversely and find the input ray parameters assoiated with a desired austi r,. Again from Eqs. (9) and (1) we obtain diretly

9 r, s fr, r r where the parameter now is. for a onvex austi, we have 1 f (1) r, i.e. From Eqs. (1) it is evident that, for r, whih ensures that the rays touhing the austi at different points do not overlap on the input plane. This allows us to invert funtion r and determine the phase r assoiated with ray harateristis r, s. Integrating Eq. (8) by introduing the new variable one gets r r r s r d r r r d (13) where is the inverse of funtion r and funtions s, r r, were obtained from Eqs. (1) by substituting for. Using Eq. (13), one an determine the phase r that must be programmed into the SLM to produe the desired austi SoR r. A harateristi ase is that of a power-law phase qr whih leads to a power-law austi also. Setting equation of the austi q r r reads n br and eliminating r from Eqs. (11), the v r f a d / f, (14) where d 1 nn 1 b 1, n 1 / n and. For n, we have 1, and hene a onvex austi SoR with a waist that starts from a maximum af to vanish on axis at 1/ at f a / d L. As was shown in [6], the point L, at whih the austi ollapses, is an infletion point for the wave amplitude, i.e. a point where the amplitude gradient along the beam axis has a loal maximum. Being very lose to the fous, this point also determines approximately the distane between the image plane and the target. Therefore the range of the beam and its maximum waist size an be adjusted through the lens foal length f, whih is one of

10 the advantages of the FT approah, as mentioned in the introdution. Equation (14) also shows that, for a given power ν, a larger b, i.e. a stronger nonlinear phase modulation, results in a smaller d, i.e. a weaker aelerating austi. This is a rather ounter-intuitive property stemming from the FT relation between objet and image waves. Also ounter-intuitively, the order of the austi v is a dereasing funtion of the order n of the nonlinear phase term; as a result, the required phase for higherpower austis 3,4,5,... is of sub-ubi order n 5 /,7 / 3,9 / 4,.... Here we would like to omment on the role of the linear phase omponent in Eq. (1). As indiated by Eqs. (6) and (14), this term is responsible for the formation of the dark disk on the image plane. Indeed, in the absene of qr, the problem redues to Fourier-transforming a wave with a linear radial phase, a situation enountered when, for example, working with Bessel beams. In this ase, no austi is formed, but rather all rays leave the objet plane in parallel and are foused by the lens on a irle on the image plane where a thin bright annulus appears (a irle for idealized Bessel beams). The inlusion of the nonlinear phase omponent subtly disturbs this perfet ray fousing, in suh a way that a smooth onvex austi is formed and the AAF phenomenon is generated. The single bright annulus on the image plane then transforms to the pattern of onentri Airy rings. In onlusion, the linear term is needed to obtain a irular fousing pattern whih the disturbane of the nonlinear phase transforms to a CAB-like pattern that evolves into an AAF wave. A losedform approximation of the beam amplitude lose and exatly on the austi SoR an be obtained by a SP omputation of the Fresnel integral of Eq. (7). As happens in other families of AAF waves [6], the field near the austi is ontributed by two lose stationary points on the input plane z, whih ollapse into a single seond-order stationary point when the field is observed exatly on the austi. The result is an Airy amplitude profile. The same method an be used to find the field at the fous, whih is now ontributed by a ontinuum of points lying on a irle on the input plane. In the Appendix an outline is given of how analytial expressions for the field in different regions an be obtained. Returning to Eq. (14), it interesting to note that AAF beams with paraboli austis require the input wavefront to be modulated with a ubi phase n 3. This is not a surprise if one takes into aount our reent analytial results on the FT of CABs [7]. In this work, the FT (expressed as a Hankel transform) of the

11 CAB Ai R r, was found to behave as Bk 3 J kr k where / 3, B k is a ompliated super-gaussian envelope. From the asymptoti behaviour of Bessel funtion, it follows that, for large k, the FT behaves proportionally to 3 os kr k / 3, i.e. as a real envelope modulated with a ubi phase. Hene, from the viewpoint of present work, the paraboli body of a CAB is a by-produt of the ubi phase modulation of its spetrum, whih an be onsidered as the analogue of the same FT property of 1D Airy beams [3]. The austi SoR of Eq. (14) develops in the half-spae z f and the generated beam is AAF with its fous ourring on axis shortly after z f L. This is a result of the phase modulation in Eq. (1) being of diverging nature. If instead, the omplex onjugate initial ondition u * r is assumed (onverging phase modulation), then the entire field in the half-spae behind the lens beomes u * 4 f z, i.e. it is mirrored with respet to the foal plane and the mirrorsymmetri of Eq. (14) austi is formed in f z f. Moreover, as a result of the onverging phase modulation itself, another austi SoR is formed before the lens z f, having the expression r z dz a. This austi is asymptoti (varies as 1 z ) to the input plane and, after passing through the lens, it experienes an inward slope disontinuity and transforms into a power-law austi. If additionally the beam parameters are hosen so that L f d af, then the transmitted powerlaw austi ollapses at z f L, a point of maximum but negative amplitude gradient, thus imparting to the transmitted beam a AADF harater. In the spirit of our previous work [7], we term this ondition as the weak-hirp regime. On the other hand, when L f d af, the hirp is strong enough to make the austi ollapse before the lens at z d / a 1/ and no fous ours after the lens. As shown in [7], if the input amplitude is properly engineered, the generated CAB an have two foi, the first being AADF and the seond AAF. As a result, an elegant, perfetly mirror-symmetri optial bottle is formed between the two foi whih an be used as an optial trap. Optial bottles an also be built by the approah presented here, however with some additional effort by letting the input beams u r

12 and u * r interfere in a standing wave pattern of the form Ar r os. In that ase eah of the two omponents reates half of the full austi. The parameters should, of ourse, be tuned in the weak-hirp regime, so that the bottle lies entirely behind the lens. An illustrative example is given in the next Setion for an optial bottle with sinusoidal shape. 3. Numerial Examples To illustrate our analytial arguments, we devote this Setion to numerial simulations. Note that, in all of the following figures, the spatial oordinates are normalized. To give a sense of the beam s atual extent, typial values for the length sales an be x 5 m in the transverse and x / m in the longitudinal diretion, at a wavelength around 5 nm. Let us first demonstrate the AAF and AADF mehanisms through the ray optis piture. Figure (a) presents the results of ray traing for a wave with the phase modulation of Eq. (1) and the parameters n 3, a 1, b 1/ 3, being Fourier- transformed by a lens with f 1 (also in x / units). The hirp parameter has 3 been hosen to satisfy b f austi 1 / 4, 1/ 3, resulting in the formation of the paraboli r (indiated with a dashed urve) whih is familiar from 1D Airy beams [3]. In Fig. (b), the orresponding ray pattern is depited for a beam with the same parameters but with the omplex onjugate input amplitude. Note how the exatly symmetri austi now develops in f z f, and also the austi 5 1 developing in z f. The two urves meet at the lens plane r z z with different slopes, as a result of the lens fousing ation. Wave simulations of the two previous onfigurations are shown in Fig. 3 and 4, respetively, as obtained by numerially solving the paraxial equation of propagation. In both ases the envelope of the input beam has been assumed to be the Gaussian r r exp / 45. The results learly verify our expetations from the ray-optis approah. In Fig. 3(a), the wave initially expands due to the diverging phase modulation; however, after passing through the lens, it starts to onverge and it eventually forms an evident austi SoR beyond the image plane. The superposed

13 analytial urve of Eq. (14), drawn with a dashed line, onfirms the expeted paraboli profile of this austi. Figure 3(b) depits the maximum intensity over the transverse plane versus the propagation distane in logarithmi sale. The onset of the AAF phase is signified by an evident knee in the I max urve (indiated with an arrow) that ours at z 5.8. Shortly after, at z f L 6.3, the austi ollapses and the wave amplitude 1 / max has an infletion point, i.e. its gradient is maximum. The abruptness of fousing an be appreiated by the jump in the rate of inrease of max log I whih is averagely. (per z unit length) before and.3 after the knee. A reversed, defousing behavior is obtained in Fig. 4, where the opposite phase modulation has been applied on the same Gaussian wavefront. Now the wave before the lens onverges and forms a austi that is well fitted by the equation (5 1) (dashed line), that was predited by ray-optis. Passing through the r z z lens, this austi experienes an inward slope disontinuity and transforms into the part of the parabola 1 / 4 r for. As this austi SoR ontrats, I max, shown in Fig. 4(b), exhibits a series of maxima with gradually inreasing strength. The last maximum, loated at z 13.4, signifies the onset of the AADF phase that ontinues up 14. z with a large negative slope for log I max around -.3 in average. At this point the I max urve has a knee (indiated by an arrow), that introdues a phase of slow defousing with a slope -. for log I max. Between the maximum and the knee, an infletion point ours at z 13.7 as a result of the ollapse of the austi. The austi extends up to the foal plane, beyond whih the wave is ompletely diffrated. As another example, we wish to produe an optial bottle beam with the sinusoidal austi r af os. Parameter κ determines the length of the bottle along the z axis, i.e. approximately /. Inserting this equation into Eq. (13) and ompleting the algebra, the required phase is af 1 r x 1 sin x 3x 1 x, 4 (15) where x r / af is a normalized radius. Expression (15) is valid for x 1, i.e. for points on the input plane with r af. Rays emitted from that disk reate the half-

14 period of the sinusoidal austi /. For, r af r is hosen so that the austi is ontinued smoothly for /. Among infinite possibilities, we here opt for a paraboli ontinuation, whih is expressed by af r / 4 4. Substituting again into Eq. (13), the orresponding extension of the phase is found af 3 r x 6x 1, x 1. (16) 4 Figure 5 shows the simulation results for the parameters / 1, a 1, f 1. The input amplitude is the standing wave u r exp[ r / 45 ]os r that is needed to form the omplete austi. The results verify the formation of the optial bottle with a shape that agrees well with the superposed sinusoidal urve (Fig. 5(a)). In the Imax z urve of Fig. 5(b), the AADF and AAF phases defining the ends of the bottle are evident. Notie also in the inset of the same figure the harateristi Airylike pattern of onentri rings developing on the image plane. We onlude by noting that the proposed FT method is perfetly appliable for produing aelerating 1D or D beams with pre-engineered trajetories. Indeed, the results of Setion are valid for (1+1)D beams evolving in Cartesian oordinates x-z and being Fourier transformed by a ylindrial lens. Then, in analogy to Eq. (1), the 1D input ondition is u x Ax i x reads exp, where the phase modulation now x ax qx, (17) with x running from to. For a given funtion q, the equation of the 1D austi is again found from Eq. (11), replaing r with x and r with x, and s x x in analogy to Eq. (8). Inversely, for a desired austi x, the required phase is found from Eq. (13) with the same substitutions. For example, one finds that, in order to produe the power-law austi of Eq. (14), the nonlinear phase n term should be qx sgn x b x, where sgn x is the sign funtion. The linear term ax an now be freely tuned, and even be negative or zero, to shift the beam

15 laterally. An example is shown in Fig. 6(b) for a onfiguration with the parameters n 5 /, a, b.53, f 1 and the Gaussian envelope x x exp[ / 45 ]. The resulting 1D beam trajetory agrees very well with the superposed third-order austi 3 r / 7. Note also in the orresponding ray traing result of Fig. 6(b) that the two symmetri parts of the austi, before and after the image plane, are formed by rays emitted from half-planes x and x of the input wavefront, respetively. Using separation of variables, the above an be readily generalized in D beams. For example, the beam with input amplitude u x, y A x A y exp i x i y, will produe a austi that aelerates along the x y diretion. An example is shown in Fig. 7, for the envelope and phase funtions of the previous example. 4. Conlusions We have proposed a simple yet general method for produing in the Fourier spae AAF and AADF optial beams with arbitrary onvex austi SoR. The method involves the radial modulation of a simple optial wavefront with an appropriately designed nonlinear phase struture and subsequently a FT operation by means of a lens. The required phase is the sum of a linear and a nonlinear term, with the seond being responsible for the shape of the austi. Through both wave and ray optis we showed how arbitrary power-law austis an result from a power-law nonlinear phase omponent. In partiular for seond-order austis, we realized that the paraboli shape is a result of the ubi phase modulation of the beam s spetrum, a onlusion that we have previously reahed speifially for CABs. The FT method proposed here is expeted to offer ertain advantages over the diret generation of AAF beams using phase masks only, suh as the ability to easily ontrol the size and range of the produed beams through the lens foal strength. In addition perfetly symmetri optial bottles an be reated and saled at will. Our proedure an also be used for produing 1D or D beams with arbitrary onvex austis, thus providing a versatile tool for all kinds of aelerating waves.

16 5. Aknowledgement This work was supported by the FP7-REGPOT-9-1 projet Arhimedes Center for Modeling, Analysis and Computation (ACMAC), by the United States Air Fore Offie of Sientifi Researh (USAFOSR-grant nos. FA and FA ) and by the National Siene Foundation (PHY-897). 6. Appendix Here we briefly outline how the field amplitude of the AAF wave an be approximately derived in analyti form. Referring to Fig. a, the field beyond the image plane is written 1 r r ur, u, exp i J d, i (A1) where the field on the image plane is given by Eq. (). As implied by Eq. 7, the latter field an be expressed as an envelope modulated with a nonlinear phase iq u, exp, (A) with the negative sign of the phase indiating fousing. For AAF waves with powerlaw austis, the phase is of the form Q af, with 1, as an be dedued from Eq. 7. Sine the above integral annot be omputed analytially, a SP approximation is usually employed. From the ray traing result of Fig. a it an be shown that one must distinguish between four regions whih are defined from the austi SoR and the ylinder r af. The regions are shown in Fig. 8. In region #1, whih lies in r af and outside the austi SoR, the field at eah point is ontributed by a single ray. In region #, whih lies in r rays meet at eah point. In region #3, whih lies in r af and outside the austi SoR, two af and inside the austi SoR, three rays meet at eah point one oming from the right and two from the left halfplane. Finally, in region #4, lying in r af and inside the austi SoR, four rays two from either side ontribute to the field observed at eah point. The above are valid off axis. On axis a ontinuum of rays, emerging from a irle on fous or two

17 irles off fous, interset at eah point and a different approah is appliable, as explained in the following. Now let us see what is the appropriate numerial treatment for eah region. In regions #1, #3 where r af, the argument of J in Eq. A1 is large enough to yield osillations. Then one substitutes Eq. 4 into Eq. A1, also using also used Eq. A, to obtain the double integral 1 r ros ur, exp i iq dd, i (A3) whih is subsequently treated with the SP method for two variables. The stationarity and,, onditions Q r /, three solutions,,, in region #3. 1, 3 have one solution, in region #1 and The above approah an also be applied in regions # and #4, provided that the point is not too lose to the axis. Note also that, in region #, as the austi is approahed before its ollapse, the two first-order stationary points merge to a seond-order one and a third-order expansion of the phase leads to the familiar Airy dependene of the field amplitude the fold using terminology of atastrophe theory). Note that suh an expansion is always possible sine Q af 3 with, hene the third derivative of the phase is never zero. Finally, for points near and exatly on axis, the Bessel funtion fator in Eq. A1 varies slowly and an be absorbed in the envelope. Then one an apply the standard SP method to the integral 1 r r ur, J exp i iq d, i (A4) as was done in [6]. The stationarity ondition Q / yields two first-order stationary points whih merge into a seond-order one as the ollapse point the austi is approahed from above. By third-order expanding the phase, one again finds that the on-axis amplitude near the fous is proportional to an Airy funtion. 1

18 Referenes 1. N. K. Efremidis, D. N. Christodoulides, Abruptly autofousing waves, Opt. Lett., vol. 35, no. 3, pp , 1.. G. A. Siviloglou and D. N. Christodoulides, Aelerating finite energy Airy beams, Opt. Lett. 3, (7). 3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Observation of Aelerating Airy Beams, Opt. Lett., vol. 3, pp (7). 4. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Observation of abruptly autofousing waves, Opt. Lett. 36, (11). 5. P. Zhang, J. Prakash, Ze Zhang, M. S. Mills, N. Efremidis, D. Christodoulides, and Z. Chen, Trapping and guiding miropartiles with morphing autofousing Airy beams, Opt. Lett., vol. 36, no. 15, pp (11) 6. I. Chremmos, N. K. Efremidis, D. N. Christodoulides, Pre-engineered abruptly autofousing beams, Opt. Lett., vol. 36, no. 1, pp , I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen: Fourier-spae generation of abruptly autofousing beams and optial bottle beams, Opt. Lett., vol. 36, no. 18, pp , G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Observation of aelerating Airy beams, Opt. Lett. 3, (7). 9. E. Greenfield, M. Segev, W. Walasik, and O. Raz, Aelerating light beams along arbitrary onvex trajetories, Phys. Rev. Lett. 16, (1), 139 (11). 1. L. Froehly, F. Courvoisier, A. Mathis, M. Jaquot, L. Furfaro, R. Giust, P. A. Laourt, J. M. Dudley: Arbitrary aelerating miron-sale austi beams in two and three dimensions. Opt. Express, vol. 19, no. 17, pp , August L. Felsen, N. Maruvitz: Radiation and Sattering of Waves, (Wiley-IEEE Press 1994)

19 Figures Figure 1: (Color online) Shemati of a ray travelling through a single-lens FT system. The lens is positioned at z f. The red (onvex) urve indiates the austi formed beyond z f.

20 (a)

21 (b) Figure : (Color online) Ray patterns on a vertial plane for a beam with input phase: a) r brn and b) r br n, with parameters n 3, a 1, b 1/ 3, and a lens with f 1. The red urves (ray envelopes) are the austis, while the dashed horizontal lines indiate the lens plane z f and the image plane z f.

22 r 1 4 I max (a) (b) z Figure 3: (Color online) (a) Amplitude evolution of a beam with Gaussian input envelope with e -1 radius w 45 and the phase modulation parameters of Fig. (a). The dashed urves are the austis of Eq. (11). (b) Maximum intensity versus propagation distane. The arrow indiates the point where the AAF starts. The dashed vertial lines indiate planes z f and z f.

23 r 1 4 I max (a) (b) z Figure 4: (Color online) (a) Amplitude evolution of a beam with the same parameters with Fig. 3 but with the omplex onjugate input amplitude. The dashed urves indiate the austis. (b) Maximum intensity versus propagation distane. The arrow indiates the point where the AADF ends. The dashed vertial lines indiate planes z f and z f.

24 r 1 3 (a) (b) I max z Figure 5: (Color online) (a) Amplitude evolution of a beam with input amplitude r r exp[ / 45 ]os, where is given by Eqs. (14),(15) for /1, a 1, f 1. The dashed urves indiate the sinusoidal part of the austi. (b) Maximum intensity versus propagation distane. The dashed vertial lines indiate planes z f and z f. The inset shows the amplitude distribution on the image plane.

25 35 (a) 35 (b) z x - x Figure 6: (Color online) (a) Amplitude evolution of a 1D beam with parameters n 5 /, a, b.53, a Gaussian input envelope with e -1 width 9 and a lens with f 1. The envelope urve is the austi, while the dashed horizontal lines indiate the lens and image planes. (b) Corresponding ray pattern.

26 y 1-1 (a) (b) () - - x x x Figure 7: (Color online) Transverse amplitude at different z-planes of an aelerating D beam produed by Fourier-transforming the input wavefront with u u u x, y A x A y exp i x i y, u 5/ exp[ / 45 ] and.53 sgn u u. (a) z=, (b) z=3, () z=6. The lens has f 1.

27 Figure 8: Regions for ray optis omputations of the field of an AAF wave. The regions are haraterized by the number of rays ontributing to the field at any point.