Accelerating nondiffracting beams

Physics Letters A 379 (2015) 983–987

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Physics Letters A www.elsevier.com/locate/pla

Accelerating nondiffracting beams Shaohui Yan, Manman Li, Baoli Yao ∗ , Xianghua Yu, Ming Lei, Dan Dan, Yanlong Yang, Junwei Min, Tong Peng State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China

a r t i c l e

i n f o

Article history: Received 3 December 2014 Received in revised form 19 January 2015 Accepted 20 January 2015 Available online 22 January 2015 Communicated by V.A. Markel Keywords: Accelerating beams Bessel beams Mathieu beams Parabolic nondiffracting beams

a b s t r a c t We present a set of beams which combine the properties of accelerating beams and (conventional) diffraction-free beams. These beams can travel along a desired trajectory while keeping an approximately invariant transverse profile, which may be (higher-order) Bessel-, Mathieu- or parabolic-nondiffractinglike beams, depending on the initial complex amplitude distribution. A possible application of these beams presented here may be found in optical trapping field. For example, a higher-order Bessel-like beam, which has a hollow (transverse) pattern, is suitable for guiding low-refractive-index or metal particles along a curve. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In 1979, Berry and Balazs [1] predicted a new family of solutions to the Schrodinger equation for a free particle: Airy wave packets. Airy wave packets have unique propagation behaviors of non-diffraction and free acceleration. In 2007, Siviloglou et al. predicted theoretically [2] and demonstrated experimentally [3] finite energy Airy wave packets in the optics domain. These optical Airy wave packets, called Airy beams, exhibit the properties of quasidiffraction-free and free acceleration along a parabolic curve over a certain distance. Since then, related applications including optical manipulation and generation of curved plasma channel [4–6] were also demonstrated. In parallel, various methods are also proposed to generate Airy beams, such as using a spatial light modulator (SLM) [2,4], a continuous transparent phase mask [6], asymmetric nonlinear photonic crystals [7] and surface plasmon polariton fabrication [8]. Usually, for an accelerating beam along a parabolic curve, the transverse profile of the beam is described by the Airy function modulated by a decay factor. However, Bandres [9] found that, under the condition of parabolic acceleration, the equation governing the transverse profile of beam allows for the separation of variables in the parabolic coordinate system. Thus, accelerating parabolic beams are also possible. The fields of such beams exhibit well defined parabolic nodal lines.

*

Corresponding author. E-mail address: [email protected] (B. Yao).

http://dx.doi.org/10.1016/j.physleta.2015.01.017 0375-9601/© 2015 Elsevier B.V. All rights reserved.

Recently, beams with non-parabolic trajectories were also demonstrated in both paraxial regime [10,11] and non-paraxial regime [12–17]. With the method of stationary phase, Greenfield et al. [10] created one-dimensional accelerating beams along arbitrary convex trajectories. The transverse profile of such beams is described by an Airy-like function. In the paraxial condition, the trajectory of an accelerating beam is limited to a small angle, that is, it cannot bend to large angles at which the beam is no longer shape preserving. To overcome this restriction, non-paraxial accelerating beams (NABs) have been identified theoretically and demonstrated experimentally. In two-dimensional case, NABs can be exact solutions of the Helmholtz equation (HE) in different cylindrical coordinate systems [14,15]. Unlike paraxial accelerating beams, which finally break down at large distance, NABs can travel along different planar curves beyond the paraxial limit including circle [14], ellipse and parabola [15]. Moreover, they can bend themselves to a large angle close to 90◦ — perpendicular to the original direction of propagation. For circular accelerating beams, the dynamic properties of acceleration and diffraction-free occur simultaneously. More recently, circular accelerating beams were generalized to three-dimensional case in the spherical and (prolate) spheroidal coordinate systems [16], and the parabolic coordinate system [17]. Classified by the indices of some special functions, these three-dimensional circular accelerating beams exhibit different transverse modes (in the azimuthal plane). For example, in the spherical coordinated system, the transverse mode is specified by the indices of the associated Legendre function.

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In general, the transverse profile of accelerating beams shows an asymmetry. Recently, Chremmos et al. [18] found a set of accelerating beams which exhibit a zeroth-order Bessel-like property in their transverse profile. This theoretical work was later proved by Zhao et al. [19]. In this paper, we extend the work of Chremmos et al. to beams that have a transverse profile resembling conventional diffractionfree (higher-order) Bessel beams, Mathieu beams or parabolic nondiffracting beams, depending on the initial complex amplitude distribution. We know that these three types of beams are diffractionfree solutions in the cylindrical [20], elliptic cylindrical [21,22] and parabolic cylindrical [23] coordinate systems respectively, which preserve their shapes during propagation. 2. Theoretic consideration The previous work of Chremmos et al. gave a set of zerothorder Bessel-like accelerating beams with arbitrary trajectories. The key point of the method used there is to find an initial field distribution in the input plane Z = 0





u (x, y ) = A (x, y ) exp i Q (x, y ) ,

(1)

whose paraxial Fresnel integral

u( X , Y , Z ) =

¨

1

u (x, y )e i

i2π Z

( X −x)2 +(Y − y )2 2Z

dxdy

(2)

will give a field distribution in the region Z > 0 having a desired accelerating trajectory given by X = f ( Z ) and Y = g ( Z ). As in [18], the transverse and longitudinal coordinates are divided by X 0 and k( X 0 )2 respectively, where k is the wave number and X 0 an arbitrary constant. In [18], only the initial phase Q (x, y ) is concerned. To avoid confusion, we denote by Q 0 (x, y ) the initial phase used there. By the stationary phase conditions ∂x Q 0 = 0 and ∂ y Q 0 = 0 on the curve, and the integrable condition ∂x ∂ y Q 0 = ∂ y ∂x Q 0 , Chremmos et al. found that the stationary phase points contributing to the point ( f ( Z ), g ( Z ), Z ) on the accelerating curve consist of a circle C ( Z ) described by

(x − xc )2 + ( y − y c )2 = R 2 , Z f  , yc

(3) Z g  with the prime denoting the

where xc = f − =g− derivative relative to Z , and R is a function of Z . Solving this equation for Z in favor of x and y, and integrating the stationary phase condition ∂x Q 0 = 0 or ∂ y Q 0 = 0, the initial phase Q 0 (x, y ) is found to be

Q 0 (x, y ) =

1

ˆZ

  2 f

2

  2 + g  − ( R /s)2 ds

0

  − ( f − x)2 + ( g − y )2 /(2 Z ).

(4)

To see that the initial phase distribution Q 0 (x, y ) gives a zerothorder Bessel-like accelerating beam, note that for a given Z , the field in the transverse neighborhood of the point ( f ( Z ), g ( Z ), Z ) is mainly contributed by the circle C ( Z ) in the initial plane. Chremmos et al. [18] showed that the rays emanated from the points on the circle C ( Z ) interferes constructively to give a field pattern as





u (δ X , δ Y , Z ) = exp i P ( Z )

˛

×





exp i ( f − x)δ X + i ( g − y )δ Y dl,

(5)

C(Z)

where P ( Z ) is a function of Z [18]. It is recognized that, the righthand side of (5) is the integral representation of a zeroth-order

Bessel function. To yield a higher-order Bessel function, we simply multiply the integrand in the integral in Eq. (5) by a modulation factor h(φ) = exp(imφ), which gives a Bessel function of order m. Furthermore, if we put h(φ) to be the eigen functions appearing in the integral representation of Mathieu beams or parabolic nondiffracting beams, then the field obtained from (4) will exhibit an elliptic or parabolic geometry. Noting that the integral in (5) is performed over a certain circle C ( Z ), the introduction of the factor h(φ) is also confined to this circle, that is, its argument φ is a function of Z . Since the function h(φ) is complex valued, there results an additional phase Arg(h(φ)) besides Q 0 (x, y ) given by Eq. (4). Then, the final initial phase and magnitude occurring in Eq. (1) are Q (x, y ) = Q 0 (x, y ) + Arg(h(φ( Z ))) and A (x, y ) = |h(φ( Z ))|, where the function Z (x, y ) is obtained by solving Eq. (3). We emphasize that by the introduction of the modulation factor h(φ), the stationary phase condition ∂x Q = 0 and ∂ y Q = 0 may not hold on the accelerating curve, implying that the maxima of field may not occur on the curve. However, this is what we expect, since, for example, a higher-order Bessel-like (accelerating) beam must have a transverse profile of donut shape. As pointed out by Chremmos et al. [18], the accelerating beams obtained by this method exhibit a well-defined acceleration behavior only for Z ≤ Z m , where Z m is a critical value determined by Eq. (3). For Q (x, y ) to be well defined, as in [18], the trajectory beyond Z m is set to be a straight line tangent to the well defined acceleration curve at the ultimate point ( f ( Z m ), Z m ). 3. Simulations and discussions In what follows, we confine the acceleration trajectory to a curve in the Y = 0 plane (g = 0) and put R ( Z ) = Z in Eq. (3). Using the above method, we obtain the initial field distribution u (x, y ) and substitute it into the Fresnel integral (2) to carry out numerical simulations. We first give an example of the higher-order (m = 2) version of the zeroth-order Bessel-like beam along the parabola f = Z 2 /40 given in Fig. 2 of [18]. As in [18], the initial amplitude is modulated by the Gaussian factor exp[−(x2 + y 2 )/900]. Noting that here (x, y ) and ( X , Y , Z ) are dimensionless coordinates corresponding to the actual coordinate variables (x X 0 , y X 0 ) and ( X X 0 , Y X 0 , k Z ( X 0 )2 ) with k being the wave number and X 0 an arbitrary constant. If X 0 = 100 μm and the wavelength λ = 1 μm, an 80 × 80 transverse region such as Fig. 1(a) would correspond to an actual region of 8 mm × 8 mm, while the longitudinal interval [0, 30] such as Fig. 1(c) would correspond to [0, 188.4 cm]. The phase Q (x, y ) and the magnitude A (x, y ) are given in Figs. 1(a) and 1(b) where the magnitude A (x, y ) is just the Gaussian factor. Compared to the phase distribution of the zeroth-order Bessel-like beam given in [18], the phase here shows some similarities in the outer region. In the central region, however, Fig. 1(a) presents a twisted structure, a typical property of phase vortex. This is reasonable, since on each circle C ( Z ), a local vortex phase factor exp(imφ) exists. Fig. 1(c) describes the dynamics of propagation of the beam in the Y = 0 plane, where, as desired, a parabola trajectory (dashed line) is observed. We note that the whole field distribution in the Y = 0 plane consists mainly of two curved strips located on two sides of the prescribed parabola trajectory. Figs. 1(d)–1(f) show the transverse profiles at Z = 5, 15 and 25, respectively, exhibiting the multiple circular ringed structure with a vanishing field at the center, the typical pattern of a higherorder Bessel beam. This transverse profile, in combination with the parabola trajectory, enables us to obtain a curved donut beam in three-dimensional physical space. From Figs. 1(d)–1(f), we also observe that the transverse profile is approximately invariant. However, we note that the main (central) ring is not uniform. At Z = 5, the main ring assumes its maximum roughly at the ten clock position, while at Z = 15 and 25, they are approximately at the one

S. Yan et al. / Physics Letters A 379 (2015) 983–987

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Fig. 1. (Color online.) (a) Initial phase and (b) magnitude for the second-order (m = 2) Bessel-like accelerating beam with trajectory f ( Z ) = Z 2 /40. (c) Dynamics of propagation in the Y = 0 plane. (d)–(f): Transverse intensity patterns at different Z planes.

Fig. 2. (Color online.) (a) Initial phase and (b) magnitude for the lowest-order Mathieu-like accelerating beam with trajectory f ( Z ) = (1/60) Z ( Z − 30). (c) Dynamics of propagation in the Y = 0 plane. (d)–(f): Transverse intensity patterns at different Z planes.

clock and five clock positions, respectively. These imply a rotation of the beam’s transverse pattern during propagation. We now turn to Mathieu-like accelerating beams, where the h(φ) factor is a linear combination of the Mathieu functions: h(φ) = aCem (φ, q) + bSem (φ, q), where Cem (m ≥ 0) and Sem (m ≥ 1) are the Mathieu functions, and q a parameter. We first examine the lowest-order mode where h(φ) = Ce0 (φ, q) with m = 0 and q = 2. The accelerating trajectory is assumed to be f ( Z ) = (1/60) Z ( Z − 30), designed to cross the Z axis at Z = 30. The initial phase Q (x, y ) and magnitude A (x, y ) are shown in Figs. 2(a) and 2(b), where the Gaussian factor is exp[−(x2 + y 2 )/2400]. The propagation behavior in the Y = 0 plane is plotted in Fig. 2(c), where we see that the trajectory formed by the intensity maxima at different Z is in a good agreement with the desired trajectory (dashed line). Note that in Fig. 2(c) there are no side lobes

in addition to a main lobe, showing a difference from Fig. 1(c). Figs. 2(d)–2(f) show the transverse profiles at Z = 6, 18 and 30, respectively. An intensity spot of elliptic shape is surrounded by a series of elliptic rings, a typical pattern of the lowest-order Mathieu beam [21]. These patterns remain approximately invariant during propagation. Also, note that these elliptic rings are incomplete, tending to vanish when approaching the horizontal axis. This answers why only one lobe exists in Fig. 2(c). The Mathieu analogy of higher-order Bessel beams can be obtained by setting h(φ) = Cem (φ, q) + iSem (φ, q) with m ≥ 1. In Figs. 3(a) and 3(b), the initial phase and magnitude for the Mathieu-like accelerating beam with m = 3 and q = 2 are plotted. The accelerating trajectory and the Gaussian factor are the same as in Fig. 2. Again, a vortex-like structure is observed in Fig. 3(a). From the propagation dynamics shown in Fig. 3(c), a two-strip

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Fig. 3. (Color online.) (a) Initial phase and (b) magnitude for the third-order (m = 3) Mathieu-like accelerating beam with trajectory f ( Z ) = (1/60) Z ( Z − 30). (c) Dynamics of propagation in the Y = 0 plane. (d)–(f): Transverse intensity patterns at different Z planes.

Fig. 4. (Color online.) (a) Initial phase and (b) magnitude for the parabolic-nondiffracting-like accelerating beam with trajectory f ( Z ) = −2 exp(−0.08Z ). (c) Dynamics of propagation in the Y = 0 plane. (d)–(f): Transverse intensity patterns at different Z planes.

pattern similar to Fig. 1(c) is found, suggesting a hollow-like structure of transverse profile. This is verified in Figs. 3(d)–3(f), which give the transverse profiles of the beam at Z = 6, 18 and 30, respectively. These patterns resemble those of Fig. 1 with the circular rings replaced by elliptic rings. We next examine the accelerating version of conventional parabolic nondiffracting beams. They are diffraction-free solutions in the parabolic cylindrical coordinate system. Their spectrum functions have even modes as A e (φ, q) = |sin(φ)|−1/2 × exp(iq tan(|φ/2|)) and odd modes as A o (φ, q) = i A e for −π < φ < 0 and A o (φ, q) = −i A e for 0 < φ < π [23]. Here q is a parameter related to the separation constant. Here, we choose the h(φ) factor to be h(φ) = A e (φ, q) + i A o (φ, q). With this choice, the corresponding transverse mode will consist of a series of parabola curves (a pure A e or A o mode leads to nodal lines ap-

pearing in the pattern). For the accelerating curve, we choose f ( Z ) = −2 exp(−0.08Z ). In Figs. 4(a) and 4(b), the phase and the magnitude in the initial plane are plotted, where q = 2 and the Gaussian factor is exp[−(x2 + y 2 )/2400]. We see that the magnitude in Fig. 4(b) is completely asymmetrical, where the magnitude in the y < 0 region vanishes. This arises from the fact that the amplitude factor h(φ) [= A e (φ, q) + i A o (φ, q)] vanishes for φ < 0 due to the definition of A o (φ, q). In Fig. 4(c), the intensity maxima at different Z are seen to coincide with the prescribed curve. Unlike above, the amplitude evolution here exhibits an asymmetry. Above the accelerating curve (dashed line), there are a set of side lobes which decay when moving away from the curve, while below the curve the field almost vanishes. The transverse profiles shown in Figs. 4(d)–4(f) demonstrate what we expect: a transverse pattern consist of a series of (approximate) parabola curves. Again, we find

S. Yan et al. / Physics Letters A 379 (2015) 983–987

that the transverse profiles experience rotation with increasing the value of Z . 4. Conclusion In conclusion, we have presented a set of beams that combine the properties of accelerating beams and (conventional) diffraction-free beams. The obtained beams undergo acceleration during propagation while keeping approximately invariant transverse profiles resembling those of (conventional) diffraction-free beams such as higher-order Bessel beams, Mathieu beams and parabolic non-diffracting beams. The method of obtaining these beams is based on the technique given by Chremmos et al. [18], who used the stationary phase method to obtain the phase distribution of generating a zeroth-order Bessel-like beam with a prescribed curve. With this phase, additional (complex) amplitude corresponding to a conventional diffraction-free beam is imposed to obtain the desired beam. A possible application of the beams presented here may be found in optical trapping field. For example, Airy beams can be used to guide micro particles along a parabola curve [4]. For particles with low index of refraction or metal particles, the Airy beam may fail to guide. This comes from the fact that the Airy beam attains a maximum at the main lobe, where particles of low index of refraction or metal particles tend to escape. However, with a higher-order Bessel-like beam, this problem can be overcome since the hollow (transverse) pattern is suitable for trapping these particles, while the accelerating property of the beam still acts to guide the particles. Acknowledgements This research is supported by National Basic Research Program (973 Program) of China under Grant No. 2012CB921900, and the Natural Science Foundation of China (NSFC) under Grant Nos. 61205123 and 61377008. References [1] M. Berry, N. Balazs, Nonspreading wave packets, Am. J. Phys. 47 (1979) 264–267. [2] G.A. Siviloglou, D.N. Christodoulides, Accelerating finite energy Airy beams, Opt. Lett. 32 (2007) 979–981.

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[3] G. Siviloglou, J. Broky, A. Dogariu, D. Christodoulides, Observation of accelerating Airy beams, Phys. Rev. Lett. 99 (2007) 213901. [4] J. Baumgartl, M. Mazilu, K. Dholakia, Optically mediated particle clearing using Airy wavepackets, Nat. Photonics 2 (2008) 675–678. [5] P. Zhang, J. Prakash, Z. Zhang, M.S. Mills, N.K. Efremidis, D.N. Christodoulides, Z. Chen, Trapping and guiding micro-particles with morphing auto-focusing Airy beams, Opt. Lett. 36 (2011) 2883–2885. [6] P. Polynkin, M. Kolesik, J.V. Moloney, G.A. Siviloglou, D.N. Christodoulides, Curved plasma channel generation using ultraintense Airy beams, Science 324 (2009) 229–232. [7] T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, A. Arie, Nonlinear generation and manipulation of Airy beams, Nat. Photonics 3 (2009) 395–398. [8] L. Li, T. Li, S. Wang, C. Zhang, S. Zhu, Plasmonic Airy beam generated by inplane diffraction, Phys. Rev. Lett. 107 (2011) 126804. [9] M.A. Bandres, Accelerating parabolic beams, Opt. Lett. 33 (2008) 1678–1680. [10] E. Greenfield, M. Segev, W. Walasik, O. Raz, Accelerating light beams along arbitrary convex trajectories, Phys. Rev. Lett. 106 (2011) 213902. [11] L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. Lacourt, J. Dudley, Arbitrary accelerating micron-scale caustic beams in two and three dimensions, Opt. Express 19 (2011) 16455–16465. [12] A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, J.M. Dudley, Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light, Opt. Lett. 38 (2013) 2218–2220. [13] F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P.-A. Lacourt, M. Jacquot, J.M. Dudley, Sending femtosecond pulses in circles: highly nonparaxial accelerating beams, Opt. Lett. 37 (2012) 1736–1738. [14] I. Kaminer, R. Bekenstein, J. Nemirovsky, M. Segev, Nondiffracting accelerating wave packets of Maxwell’s equations, Phys. Rev. Lett. 108 (2012) 163901. [15] P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, X. Zhang, Nonparaxial Mathieu and Weber accelerating beams, Phys. Rev. Lett. 109 (2012) 193901. [16] P. Aleahmad, M.-A. Miri, M.S. Mills, I. Kaminer, M. Segev, D.N. Christodoulides, Fully vectorial accelerating diffraction-free Helmholtz beams, Phys. Rev. Lett. 109 (2012) 203902. [17] M.A. Bandres, M.A. Alonso, I. Kaminer, M. Segev, Three-dimensional accelerating electromagnetic waves, Opt. Express 21 (2013) 13917–13929. [18] I.D. Chremmos, Z. Chen, D.N. Christodoulides, N.K. Efremidis, Bessel-like optical beams with arbitrary trajectories, Opt. Lett. 37 (2012) 5003–5005. [19] J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I.D. Chremmos, N.K. Efremidis, D.N. Christodoulides, Z. Chen, Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories, Opt. Lett. 38 (2013) 498–500. [20] J. Durnin, Exact solutions for nondiffracting beams. I. The scalar theory, J. Opt. Soc. Am. A 4 (1987) 651–654. [21] J.C. Gutiérrez-Vega, M. Iturbe-Castillo, S. Chávez-Cerda, Alternative formulation for invariant optical fields: Mathieu beams, Opt. Lett. 25 (2000) 1493–1495. [22] S. Chávez-Cerda, J. Gutiérrez-Vega, G. New, Elliptic vortices of electromagnetic wave fields, Opt. Lett. 26 (2001) 1803–1805. [23] M.A. Bandres, J.C. Gutiérrez-Vega, S. Chávez-Cerda, Parabolic nondiffracting optical wave fields, Opt. Lett. 29 (2004) 44–46.