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Non-diffracting beams have been a hot topic since the Airy wave packet was introduced to optics domain from quantum mechanics. Great efforts have been made to study this theme in recent years. The researches have ranged from paraxial regime to non-paraxial regime, and a series of new non-diffracting beams have been discovered. However, most of these beams are obtained under the time harmonic condition. To break this limitation, we propose a concept of time-dependent Bessel beam in this paper, which generalizes the non-diffracting beams to non-time-harmonic regime.We start from Maxwell's equations in vacuum under non-paraxial condition using the method borrowed from the half-Bessel beam. To obtain the non-time-harmonic solution, the fourth dimensional imaginary coordinate is introduced, which refers to the covariance in the theory of special relativity. By solving the wave equation without the time harmonic condition, we obtain the analytical expression for a time-dependent beam in the form of Bessel functions. Thus we call it time-dependent Bessel beam.The diffraction properties and space-time characteristics of the time-dependent Bessel beam are investigated theoretically. The transverse intensity and the intensity distribution of the beam are calculated and discussed in detail. The wave function of the time-dependent Bessel beam is in the same form as the normal Bessel beam so that it can exhibit non-diffraction in the four dimensional space-time. When propagating along a space-time hyperbolic trajectory, the intensity of the time-dependent Bessel beam remains constant and the width of the beam decreases with propagating distance and time increasing. Besides, we deduce the critical condition of the spatiotemporal characteristics of the beam, and the result agrees well with the concept of the light cone in the theory of special relativity.The method to deduce the time-dependent Bessel beam used in this paper is universal, and it will provide a valuable access to other solutions for the wave equations under different conditions. We extend the study of non-diffracting beams from time harmonic regime to non-time-harmonic regime. Furthermore, our work demonstrates the relation between the non-diffracting accelerating beams and the theory of special relativity. We believe this work will open up a new vista and give a new insight into the research of non-diffracting accelerating beams or other relevant research fields.
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Keywords:
- time-dependent Bessel beams /
- non-diffraction /
- space-time characters /
- critical condition
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[1] Berry M V, Balazs N L 1979 Am. J. Phys. 47 264
[2] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901
[3] Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979
[4] Broky J, Siviloglou G A, Dogariu A, Christodoulides D N 2008 Opt. Express 16 12880
[5] Yue Y Y, Xiao H, Wang Z X, Wu M 2013 Acta Phys. Sin. 62 044205 (in Chinese) [乐阳阳, 肖寒, 王子潇, 吴敏 2013 62 044205]
[6] Li L, Li T, Wang S M, Zhang C, Zhu S N 2011 Phys. Rev. Lett. 107 126804
[7] Baumgartl J, Mazilu M, Dholakia K 2008 Nat. Photon. 2 675
[8] Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N, Chen Z 2011 Opt. Lett. 36 2883
[9] Polynkin P, Kolesik M, Moloney J V, Siviloglou G A, Christodoulides D N 2009 Science 324 229
[10] Li J X, Zang W P, Tian J G 2010 Opt. Express 18 7300
[11] Wang G H, Wang X F, Dong K G 2012 Acta Phys. Sin. 61 165201 (in Chinese) [王广辉, 王晓方, 董克攻 2012 61 165201]
[12] Guo C S, Wang S Z, Rong Z Y, Sha B 2013 Acta Phys. Sin. 62 084201 (in Chinese) [国承山, 王淑贞, 荣振宇, 沙贝 2013 62 084201]
[13] Hong X H, Yang B, Zhang C, Qin Y Q, Zhu Y Y 2014 Phys. Rev. Lett. 113 163902
[14] Lumer Y, Drori L, Hazan Y, Segev M 2015 Phys. Rev. Lett. 115 013901
[15] Bloch N V, Lereah Y, Lilach Y, Gover A, Arie A 2013 Nature 494 331
[16] Kaminer I, Bekenstein R, Nemirovsky J, Segev M 2012 Phys. Rev. Lett. 108 163901
[17] Zhang P, Hu Y, Li T, Cannan D, Yin X B, Morandotti R, Chen Z G, Zhang X 2012 Phys. Rev. Lett. 109 193901
[18] Durnin J, Miceli Jr J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499
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