The Self-Healing Peculiarity of Airy Beams Propagation in Free-Space

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1 IOSR Journal of Electronics and Communication Engineering (IOSR-JECE) e-issn: ,p- ISSN: Volume 11, Issue 5, Ver. I (Sep.-Oct.016), PP The Self-Healing Peculiarit of Air Beams Propagation in Free-Space Haitao Wang a, b, *, Chengu Fan a a Ke Laborator of Atmospheric Composition and Optical Radiation, Anhui Institute of Optics and Fine Mechanics, Chinese Academ of Sciences, Hefei Anhui, 30031, China b State Ke Laborator of Pulsed Power Laser Technolog, Electronic Engineering Institute, Hefei 30037, China Abstract: We have theoreticall and numericall investigated the self-healing peculiarit of Air beams truncated b different deca factors. The self-healing dnamic processes of truncated Air beams are different from that of blocked Air beams. The wing with large truncation of Air beam can propagate much longer distance than that of small truncated wing. When one of the wings is complete truncated, the Air beam propagates along a straight line and neither self-bending nor self-reconstruction can be observed. The Ponting vector and energ flow are conducted to gain an insight into the phsical mechanism. Kewords: ( ) Propagation; ( ) Diffraction; ( ) Diffraction theor; (60.110) Electronmagnetic optics. I Introduction Air beams famil as newl different class of non-diffracting beams, such as Bessel, Mathieu, Weber and discrete non-diffracting beams, have attracted much attention on their propagation and generation [1-4]. Besides the non-spreading feature, the remarkable peculiarities of Air beams are that the do not propagate in a straight line path but along parabolic ballistic trajector (self-bending), transversal acceleration, self-healing or self-reconstruction. Those properties offering a wide range of potential applications in the fields including plasma wave-guiding [5], microparticle manipulation [6], generation of light bullets and curved femto-filaments [7-8], and Generation of electron Air beams [9]. Air wave packet as the solution of the paraial Schrödinger equation [10] was firstl observed b Berr and Balazs under the phsical background of quantum mechanism [11]. Theoreticall, the ideal Air beam contains infinite energ. Although Air function is not square integrabel [1], the gravit center remain invariant and the main lobe tends to accelerate along parabolic trajector during propagation in spite of without eternal potential. The Air beam can be seen as an interference pattern, similar to the Bessel beam [13-14]. Eperimentall, the first optical realization of Air beam b Siviloglou et al [15-16] is a truncation version, which carries finite energ and is generated from multipling the Air function distribution b an eponential deca factor or a Gaussian window. Recentl, some other methods, such as using nonlinear processes in photonic crstal [17], or b utilizing constructed microchirp laser [18], 3/ phase pattern [19] and so on, have been proposed for producing Air beams. Due to the intriguing propagation peculiarit of Air beam (self-curving trajector, weak diffraction and self-healing), man works have investigated and demonstrated the propagation dnamics of Air beams from paraial [0] to nonparaial approimation [1-] and from spatio-temporal 1D-D to 3D-4D [3-4]. Man others Air-tpe beams or pulses and beam arra [5-7] have also triggered much attention, from linear media to the nonlinear regime [8-9]. Among those works, the self-accelerating and self-healing evolutions of Air beams [30-31], whose main lobes are embed with a vorte or masked b an artificial presupposition particle, are interesting topics and have been widel investigated. While, if the two wings or one of them is truncated, whether or not the distinguishing processes of energ compensation and flow can occur to fulfill the self-accelerating and self-healing. This question has not been reported et and remains an open question, but it will be investigated in the following arrangement. II The theoretical description and mathematical equations We will start our analsis and get the ballistic dnamics of D Air beams from the following basic paraial Schrödinger equation [15, 30]: u(,, z) 1 u(,, z) u(,, z) i 0, (1) z k where k n0 / 0 is the wavenumber and 0 53 nm is the wavelength. The initial electric field envelope is given b: DOI: / Page

2 The self-healing peculiarit of Air beams propagation in free-space u(,, z 0) Ai( sm)ep(a ms m) m, ep a Ai Ai ep a. () The evolution pattern of accelerating Air beam in closed form can be obtained from: u,, z um sm, m m, 3 m z m z az a z az z Ai i ep i i i k0 0 1k 0 k0 k 0 3 m m z az a z z az z Ai i ep i i i, (3) k0 0 1k 0 k0 k 0 the Ai( sm ) denotes the Air function and the dimensionless transverse coordinates are s / 0 the normalized propagation distance are z / k and 0 z / k0. 0 and s / 0, = 0 =1.5 mm are normalization constants and a m, represents the deca factor acts as an eponential aperture function. Along the propagation direction z path, the ballistic trajector [3-33] of the main lobe of Air beam can be directl determined b: z m z 3 z 4k 0, (4) z mz 3 z 4k 0 here, the m, denote the initial launch angles associated with the beam coordinate ais and. It illustrates that the deflection of peak intensit from straight line is wavelength dependent [34]. The self-healing properties of Air beam partiall blocked b a finite opaque obstacle can be eplained b Babinet s principle. It has been demonstrated that the robustness of the self-healing of non-diffracting optical beams in adverse environments, such as scattering media and turbulence, are corresponding to the internal power flow described b Ponting vector and angular momentum [30]. The theoretical calculating evolutions of self-bending [35] propagation configuration and self-reconstruction as a function of propagation distance are shown in figure 1(a) and 1(b). Here, the two wings of the initial Air beam are truncated b different deca factors a = 0.01 and a = 0.. Figure 1 shows that oscillating self-reconstruction dnamics of the partiall truncated tailing of -wing can be observed along the propagation path. Another -wing graduall decas into circumambience of free space. Fig. 1. (Color online) (a) The theoretical calculating self-bending propagation configuration of truncated Air beam. (b) The self-reconstruction processes of truncated Air beam. DOI: / Page

3 The self-healing peculiarit of Air beams propagation in free-space III The numerical results and discussions 3.1. The self-bending, ballistic trajector and transversal accelerating The numerical simulation of self-bending of Air beam is shown in Fig. (a). The ellow dash-line represents the ballistic trajector of the main lobe according to the theoretical formula (4). The numerical results consistent with theoretical calculations well. Along the propagation path, the two wings of Air beam developed into different propagating scenarios. In the transverse -direction, the intensit profile indicates that the beam has undergone an important transformation from Air function to a Gaussian profile as can be seen in Fig. (b). In the transverse -direction, the configuration is absolutel different from that of transverse -direction. A self-similar Air-shaped intensit profiles, featured b slowl decaing, oscillating tails, can be observed at different positions along propagation path shown in Fig. (c). As the continuing propagation, the intensit profile will finall become into a Gaussian shape as similar as that of -wing. The transversal accelerating of the Air beam in the transversal plane including and directions independent on the truncation factors indicated in formula (4). It has been demonstrated that the accelerating trajector can be controlled with opticall induced refractive-inde gradient to achieve the goal of propagation along arbitrar trajector [36-37]. Fig.. (Color online) (a) The numericall calculated self-bending propagation configuration of truncated Air beam, the ellow dash-line represents the ballistic trajector of the main lobe according to the theoretical formula (4). The intensit profiles as a function of distance of truncated Air beam, (b) corresponding to -direction and (c) corresponding to -direction. 3.. The self-healing of air beams The self-healing propert of Air beam implies that the propagation is independent of the initial condition and is hardl sensitive to small phase fluctuation of the incident beam, resulting in the sstem develops towards a stable mode, at least if one observes its intensit profile at a finite-distance. The numericall calculated self-healing processes are shown in Fig. 3, which is in ecellent agreement with Fig. 1(b) obtained from theoretical calculation. While this self-healing processes are different from that perturbed or blocked b a particle of the main lobe of Air beam. The blocked Air beam reconstructed from an interference field of its own two-wing side lobes. From the self-healing dnamics, it can be seen that -wing propagates much longer distance compared with that of -wing, due to the former undergoes a special course of self-healing firstl before the diffraction propagation manner that -wing directl encountered. Fig. 3. (Color online) (a) The numericall calculated self-healing propagation configuration of truncated Air beam with different deca factors a = 0.01 and a = 0.. DOI: / Page

4 The self-healing peculiarit of Air beams propagation in free-space While when the deca factor is adjusted to a = 1 and keep the a not changed, an absolutel different accelerating propagation manner can be observed as shown in Fig. 4(a). The transversal acceleration and position shift of the main lobe occur onl along the -wing as can be seen in Fig. 4(b), the trajector of the main beam becomes a straight line in -direction but not curved along the cross angle 45 in the - plane enabling the self-bending trajector of Air beam. The -wing ehibits a normal diffraction propagation similar as Gaussian beam in spite of the initial intensit profile with a slight oscillating tailing seen in Fig. 4(c). Fig. 4. (Color online) (a) The numericall calculated propagation configuration of truncated Air beam with different deca factors a = 0.01 and a = 1.0. The intensit profiles as a function of distance, (b) corresponding to -direction and (c) corresponding to -direction. The red dash-line represents the ballistic trajector of the main lobe in the -z plane The pointing vector and energ flow We gain an insight into the evolution to eplain the above observed patterns emanating from numerical calculation. When a = 1.0, it means the -wing of the beam is absolutel truncated b a Gaussian function window, and onl the -wing of the beam evolution with a long oscillating tailing similar with Bessel beam. It is demonstrated that the spatial truncation function plas an important inherent influence on the etrinsic evolution of the ideal Air beam. Furthermore, we conduct the practical analsis b using the Ponting vector S and energ flow to illuminate the field configuration. In the paraial regime, S 1 i S S S u z u u u u, (5) * * z 0 40k is given b: where the 0 0 / 0 represents the impedance of free space. z and S denote the longitudinal and transverse components of the Ponting vector, respectivel. In figure 5, we plot the Ponting vector, which intuitivel demonstrates the internal energ flow during the transversal acceleration and self-healing processes, of the field at two different distances for a = 0.01 and a = 0.. At the initial propagation stage z = 50 m, onl small energ of -wing flows towards the lateral regimes along direction, resulting in rebirth of the secondar side lobes from the -wing and reformation of the newl formed coherent field in the internal cross regime. The eternal energ flows evidentl facilitate the lateral acceleration and enable the self-bending of Air beam along cross angle 45 in the - plane. As propagation continued, the reconstruction dnamic will be enhanced, i.e., after z = 110 m, the complete regeneration of -wing can be observed. While when the -wing is completel truncated as shown in figure 4, most energ flow outward along the -wing and there is not enough energ efficientl compensate to realize the reformation the -wing. Therefore, an acceleration onl occurs in the direction and a straight line propagation can be obtained. DOI: / Page S

5 The self-healing peculiarit of Air beams propagation in free-space Fig. 5. (Color online) The calculated transverse energ flow at different distances for z = 50 m; (b) z = 110 m. a = 0.01 and a = 0.. (a) IV Summar We have numericall demonstrated the self-healing peculiarit of Air beam truncated b different deca factors. The self-healing dnamic processes of truncated Air beams are different from that of blocked Air beams. The wing with large truncation of Air beam will propagate much longer distance than that of small truncated wing. While the Air beam, one of the wings is complete truncated, can propagate along a straight line and neither self-bending nor self-reconstruction can be observed. It indicates that the truncation function has important influence on the propagation trajector of Air beam. The Ponting vector and energ flow are conducted to gain an insight into the phsical mechanism. Acknowledgements The project was supported b Open Research Fund of State Ke Laborator of Pulsed Power Laser Technolog, Electronic Engineering Institute (Grant Nos: SKL013KF01 and SKL015KF03); National Natural Science Foundation of China (Grant No: ) and Dean fund of Hefei Institutes of Phsical Science, Chinese Academ of Sciences (Grant No. YZJJ01506). References [1] Migule A. Bandres, Ido Kaminer, Matthew S. Mills, B. M. Rodríguez-Lara, Elad Geenfield, Morderchai Segev and Demetrios N. Christodoulides. Accelerating optics beams, Optics & Photonic News 013, [] Miguel A Bandres and B M Rodríguez-Lara, Nondiffracting accelerating waves: Weber waves and parabolic momentum, New Journal of Phsics 15, 013, [3] M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, P. Di Trapani, Finite-energ, accelerating Bessel pulses, Optics Epress 008, 16(4), [4] Gil Porat, Ido Dolev, Omri Barlev, and Ad Arie, Air beam laser, Optics Letters 011; 36: [5] Pavel Polnkin, Miroslav Kolesik, Jerome V. Molone, Georgios A. Siviloglou, Demetrios N. Christodoulides, Curved Plasma Channel Generation Using Ultraintense Air Beams Science 009, 34, 9-3 [6] Baumgartl J, Mazilu M, Dholakia K. Opticall mediated particle clearing using Air wavepackets. Nature Photonics 008,, [7] Peeter Piksarv, Andreas Valdmann, Heli Valtna-Lukner and Peeter Saari, Ultrabroadband Air light bullets, Journal of Phsics: Conference Series 014, 497, [8] Jérôme Kasparian and Jean-Pierre Wolf, Laser Beams Take a Curve, Science 009, 34, [9] Noa Voloch-Bloch, Yossi Lereah, Yigal Lilach, Avraham Gover and Ad Arie, Generation of electron Air beams, Nature, 013, 494, [10] Ido Kaminer, Rivka Bekenstein, Jonathan Nemirovsk, and Mordechai Segev, Nondiffracting Accelerating Wave Packets of Mawell s Equations, Phsical Review Letters 01, 108(16), [11] M. V. Berr and N. L. Balazs, Non-spreading wave packet, American Journal of Phsics 1979, 47(3), [1] K.B. Oldham et al., An Atlas of Functions, Second Edition, Springer Science Business Media, LLC, 009, Chapter 56, [13] S. Chávez-Cerda, A new approach to Bessel beams, Journal of Modern Optics 1999, 46, [14] Planchon T A, Liang Gao, Milkie D E, Davidson M W, Galbraith J A, Galbraith C G, et al. Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination. Nature Methods 011; 8(5): [15] Siviloglou G A, Brokl J, Dogariu A, Christodoulides D N. Observation of accelerating Air beams. Phsical Review Letters 007; 99: [16] Siviloglou G A, Christodoulides D N. Accelerating finite energ Air beams. Optics Letters 007; 3: [17] T. Ellenbogen, N. Voloch-Bloch, A. Ganan-Padowicz, and A. Arie, Nonlinear generation and manipulation of Air beam, Nature Photonics 009, 3, [18] Berna Yaliza, Burak Solu, and Selcuk Akturk, Optical element for generation of accelerating Air beams, Journal of the Optical Societ of America A 010, 7(10), [19] Don M. Cottrell, Jeffre A. Davis, and Thomas M. Hazard, Direct generation of accelerating Air beams using a 3/ phase-onl pattern, Optics Letters 009; 34: [0] Miguel A. Bandres, Accelerating parabolic beams, Optics Letters 008, 33: Torre, Air beams beond the paraial approimation, Optics Communications 010, 83, [1] Luis Carretero, Pablo Acebal, Salvador Blaa, Celia Garcia, Antonio Fimia, Roque Madrigal, Angel Murciano, Nonparaial diffraction analsis of Air and SAir beams, Optics Epress, 009, 17(5), DOI: / Page

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