Optics Communications

Transcription

1 Optics Communications 294 (2013) Contents lists available at SciVerse ScienceDirect Optics Communications journal homepage: Propagation properties and self-reconstruction of aimuthally polaried non-diffracting beams Meng He, Ziyang Chen, Shunhong Sun, Jixiong Pu n College of Information Science and Engineering, Huaqiao University, Xiamen, Fujian , China article info Article history: Received 27 August 2012 Received in revised form 2 December 2012 Accepted 8 December 2012 Available online 9 January 2013 Keywords: Aimuthally polariation Non-diffracting beams Self-reconstruction abstract Aimuthally polaried (AP) non-diffracting beams are investigated theoretically and experimentally in this paper. We generate the AP non-diffracting beams by passing laser beams through a polariation converter and an axicon. Both the non-diffracting and the self-reconstruction phenomenon are studied theoretically and experimentally. The aimuthally polaried beams with non-diffracting and self-reconstruction properties may have wide applications in many fields such as particle trapping and manipulation. & 2013 Elsevier B.V. All rights reserved. 1. Introduction In the past several years, cylindrical vector beams (i.e., laser beams with cylindrical symmetry in polariation) have attracted much attention due to their important applications, such as optical trapping, singular optics, optical data storage, optical inspection and metrology etc. [1 12]. Aimuthally polaried beams, which are an important class of laser modes whose state of polariation is cylindrical, have been investigated in detail by many researchers [1 7]. Recently, a non-diffracting dark channel with a long DOF has been achieved by tight focusing of a double-ring-shaped aimuthally polaried beam [3]. In another study, the generation of a non-diffracting transversally polaried beam by highly focusing an aimuthally polaried beam with a high-na lens and a multibelt spiral phase hologram has been analytically demonstrated [4]. The studies demonstrate that aimuthally polaried laser beams can be applied in optical tweeers for particle trapping and manipulation [10 12]. On the other hand, increasing interest has been paid to non-diffracting beams because of their potential applications [13 25]. One example of non-diffracting beams is Bessel beams, which were introduced by Durnin [13]. Ideal forms of such beams are impossible to realie, as they would have an infinite extent and carry infinite power [13]. However, finite approximations of these beams can be realied that propagate over extended distances in a diffraction-free manner. Bessel Gauss beams were introduced as a more realistic representation of Bessel-like beams that can be achieved in actual experiments [14]. Bessel Gauss beam can be obtained by propagating Gaussian beam through an axicon [19,20]. When illuminated by a Gaussian beam with a waist sie smaller than the hard aperture of the axicon, virtually the whole input intensity is converted into an approximation to a Bessel beam. The axicon-generated beam is a close approximation to a Bessel beam over a limited propagation distance. However, the intensity of the central maximum is not constant with propagation but varies smoothly. The central maximum propagates without appreciable spreading (i.e., it is non-diffracting) over this distance. Besides the non-diffracting, self-reconstruction is another fascinating property of Bessel beams. The self-reconstruction of the ero-order Bessel beam in free space was fully demonstrated and explained by Bouchal et al. [23] in Furthermore, Tao et al. [24] investigated the self-reconstruction property of the fractional Bessel beam (FBB) for three-dimensional optical trapping applications and their experimental results showed that the FBB can overcome a block of obstacles and regenerate itself after a characteristic distance. For the fascinating property, Bessel beams are useful in optical microscopy [15], interboard optical data distribution [16], and optical trapping etc. [17]. However, to the best of our knowledge, the selfreconstruction properties of AP non-diffracting beams have not been studied so far. In this paper, we extend the aimuthally polariation beams to non-diffracting case. Analytical formulae of AP beams passingthroughanaxiconarederived. The properties of AP nondiffracting beams on propagation in free space and passing through obstacles are illustrated numerically. Some interesting conclusions are obtained. n Corresponding author. Tel.: þ address: jixiong@hqu.edu.cn (J. Pu). Fig. 1. Experimental setups for generating AP non-diffracting beams /$ - see front matter & 2013 Elsevier B.V. All rights reserved.

2 M. He et al. / Optics Communications 294 (2013) Theory It is assumed that the transverse profile of the electric field of the aimuthally polaried beam is given by the expression [6] E 0 ðþ¼a r r r2 exp w w 2, ð1þ where A is a random amplitude, w is the beam radius at ¼0. In this work, the aimuthally polaried non-diffracting beams (AP non-diffracting beams) were generated by passing the aimuthally polaried beams through an axicon. The transmittance function of the axicon can be expressed as ( exp½ikðn1þgrš, r rr tðrþ¼, 0, r 4R ð2þ Fig. 2. Theoretical intensity distribution of AP non-diffracting beams, where l¼ mm, w¼2 mm, n¼1.50, g¼0.51, ¼300 mm. The arrows represent the polariation direction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) where k¼2p/l is the wave number, l is the wave length, n is the refractive index of the axicon, and g is the taper angle of the axicon. Fig. 3. Theoretical intensity distribution of AP non-diffracting beams through polaroid polarier with different angle of, where (a) y¼01; (b) y¼451; (c) y¼901; (d) y¼1351, respectively. The other parameters are the same as in Fig. 1. The arrow represents the direction of polaroid polarier.

3 38 M. He et al. / Optics Communications 294 (2013) In the Cartesian coordinate system, the local aimuthal and radial components can be expressed by the following transformations 8 <! e r ¼ e! xcos jþ! e ysin j! e j ¼ e! ycos j! ð3þ : e xsin j: According to the Collins formula [26], the y component of aimuthally polaried beam through an axicon in free space can be expressed as Z i 2p Z 1 UrUcos j E y r,y, ¼ expðikþ l 0 0 w exp r2 exp ikðn1þgr w 2 " exp ik # r2 þr 2 exp ikrr cos jy rdrdj: ð4þ For simplicity, it is assumed that A¼1. By introducing the following formulae: exp ikrr cos jy ¼ X i l krr J l exp il jy, ð5þ cos y ¼ 1 exp iy 2 ð Þþexp ð iy Þ, ð6þ sin y ¼ i exp iy 2 ð Þexp ð iy Þ, ð7þ Z 2p 0 ( expðimjþdj ¼ 2p, m ¼ 0 0, ma0, where J l krr= denotes a Bessel function of the first kind, of order l. Eq. (4) can be further simplified as ð8þ Fig. 4. (a) Intensity distribution in r plane; (b) intensity distribution of AP non-diffracting beams within the propagation range ¼200 mm 800 mm; and (c) intensity distribution of AP non-diffracting beams within the propagation range ¼800 mm 1400 mm. The other calculation parameters are same as that in Fig. 1.

4 M. He et al. / Optics Communications 294 (2013) E y r, ¼ðkÞexp ik Z 1 r 2 r2 exp w w 2 0 ikr 2 ð Þexp cos y ikr 2 exp ikðn1þgr exp J 1 krr dr: ð9þ The cross section intensity distribution of AP non-diffracting beams can be calculated by following formula: I r, ¼ Ex r, 2 þe y r, 2 : ð11þ Similarly, the x component can be expressed as follows: ikr 2 E x r, ¼ kuexpðikþexp sin y Z 1 0 r 2 r2 ikr 2 exp exp ikðn1þgr exp w w 2 J 1 krr dr: ð10þ 3. Generation and propagation of AP non-diffracting beams Fig. 1 illustrates the experimental setups for generation AP nondiffracting beams. As shown in the figure, a Gaussian beam from a He Ne laser is expanded (f 1 ¼150 mm and f 2 ¼300 mm), and then is incident into a Radial Polariation Converter (RPC). The AP nondiffracting beam is produced by passing the AP beam through an axicon. The intensity distributions are recorded by the CCD. Fig. 5. (a) Intensity distribution in r plane; (b) intensity distribution of AP non-diffracting beams within the propagation range ¼100 mm 500 mm; and (c) intensity distribution of AP non-diffracting beams within the propagation ange ¼500 mm 700 mm. g¼1.01, the other calculation parameters are same as that in Fig. 1.

5 40 M. He et al. / Optics Communications 294 (2013) Fig. 2 shows the numerical simulation of the intensity distribution of AP non-diffracting beams. The red arrows stand for the polaried direction of the beams. To better illustrate the polaried property, we present the cross sections of intensity distribution of AP non-diffracting beams through different angle Fig. 6. Experimental intensity distribution of AP non-diffracting beams with propagation distance of ¼300 mm, the other calculation parameters are same as that in Fig. 1. (y, the angle with the horiontal direction) of polaroid polarier. The white arrows in the Fig. 3 represent the polaried direction of the polaroid with (a) y¼01; (b) y¼451; (c) y¼901; (d) y¼1351, respectively. The results of Fig. 3 verified the aimuthally polaried property of the beams. The intensity distribution of AP non-diffracting beams through different taper angles of the axicon with g¼0.51 and 1.01 are shown in Figs. 4 and 5, where (a) shows 2D plot of intensity distribution in r plane, and (b) and (c) plot the normalied transverse profiles of intensity at different propagation distances. It can be seen in Fig. 4(b) (g¼0.51) that all curves overlapped within the propagation range ¼200 mm 800 mm, indicating the non-diffracting property of the beam. The non-diffracting property disappears with distance larger than 800 mm, as shown in Fig. 4(c). From the curves in Fig. 5(b) and (c), it can be obtained that the non-diffracting range of the axicon with g¼1.01 is ¼100 mm 500 mm. It can be concluded by comparing the results of Figs. 4 and 5 that the non-diffracting range decreases with the increasing taper angle of the axicon. This result is consistent with the previous researches [19]. The experimental intensity distribution of the AP non-diffracting beams with l¼ mm, w¼2 mm, n¼1.50, g¼0.51 and ¼300 mm is presented in Fig. 6. A polaroid polarier is introduced after the axicon to detect the aimuthally polaried property of the beams. From Fig. 7,wecanseetheAPnon-diffractingbeamsthrough different angle of polaroid polarier with (a) y¼01; (b) y¼451; (c) y¼901; (d) y¼1351, respectively. These experimental observations are consistent with the theoretical results in Fig. 3. Fig. 7. Experimental intensity distribution of AP non-diffracting beams through different angle of polaroid polarier, where (a) y¼01; (b) y¼451; (c) y¼901; (d) y¼1351, respectively. The other parameters are the same as in Fig. 1. The arrow represents the direction of polaroid polarier.

6 M. He et al. / Optics Communications 294 (2013) Fig. 8. Cross section of intensity distribution at different positions of AP non-diffracting beams through the sector-shaped obstacle: theoretical (top) and experimental (bottom), where (a) ¼100 mm; (b) ¼200 mm; and (c) ¼300 mm, respectively. The dash line represents the sector-shaped obstacle. The other parameters are the same as in Fig Self-reconstruction of the AP non-diffracting beams Self-reconstruction is an unusual property of Bessel beams, and has been extensively studied over the years [23 25]. It is now understood that if an obstruction is placed within the validity region of a Bessel beam, it will self-reconstruct after some distance Z min. The distance Z min is the length of the shadow region behind the obstacle and it is approximately given as [25] Z min D tan n1 2 ð Þg D ð 2 n1 Þg, ð12þ where D is the diameter of the obstruction, n is the refractive index of the axicon, and g is the taper angle of the axicon. Furthermore, if the obstruction is placed off the optical axis, the self-reconstruction distance will become longer. To illustrate that AP non-diffracting beams also possess the ability of self-reconstruction upon encountering an obstruction, we used a sector-shaped obstacle with a¼0.2p and r¼1 mm to block the AP non-diffracting beams, where a and r were the central angle and radius of the sector-shaped obstacle. Numerical simulations of the intensity distribution at the focal plane of the axicon are performed in the top row of Fig. 8. It can be found that the ring-shaped intensity distribution is destroyed in a short propagation distance, while is reconstructed for a long propagation distance. The region blocked by the sector-shaped obstacle which is overshadowed as observed began to self-reconstruct in Fig. 8(a) at a distance of 100 mm. Approximately at a distance 200 mm from the obstruction we notice that the beam is partially reconstructed, as illustrated in Fig. 8(b). And at a distance 300 mm from the obstruction we notice a complete reconstruction of the beam, as illustrated in Fig. 8(c). The complete reconstruction intensity patterns are similar to those of non-obstructed beams in Fig. 2. If we use a circular-shaped obstacle with r¼1 mm (D¼2 mm) to block the AP non-diffracting beams, the shadow region is calculated to be Z min E229 mm after the obstruction using the conventional reconstruction Eq. (12), this Z min is larger than that obtained by sector-shaped obstacle (Z min E100 mm). Clearly, the reconstruction distance depends on the dimension of the obstruction, the lagrer the obstruction is, the longer the reconstruction distance will be. The related experimental observations of the intensity distribution of the blocked AP non-diffracting beams at three different positions are also given in the bottom row of Fig. 8. As is obvious, the non-diffracting field disturbed by the obstacle is reconstructed to its initial intensity profile in the far region. The experimental observations are well consistent with the theoretical results. The results in Fig. 8 clearly indicate the self-reconstruction property of AP non-diffraction beams. 5. Conclusions The AP non-diffracting beams have been generated and investigated experimentally and theoretically. The beam propagates without diffraction within a certain distance, depending on the taper angle of the axicon. Besides that, the self-reconstruction characteristic of AP non-diffracting beams is verified. Such AP non-diffracting beams may have potential applications in particle trapping etc. Acknowledgments This research was supported by the National Natural Science Foundations of China (Grant Nos and ). References [1] K.S. Youngworth, T.G. Brown, Optics Express 7 (2000) 77. [2] C.J.R. Sheppard, S. Saghafi, Optics Letters 24 (1999) [3] B. Tian, J. Pu, Optics Letters 36 (2011) 2014.

7 42 M. He et al. / Optics Communications 294 (2013) [4] G.H. Yuan, S.B. Wei, X.C. Yuan, Optics Letters 36 (2011) [5] X. Hao, C. Kuang, T. Wang, X. Liu, Optics Letters 35 (2010) [6] D.P. Brown, A.K. Spilman, T.G. Brwon, R. Borghi, S.N. Volkov, E. Wolf, Optics Communications 281 (2008) [7] Q. Zhan, Advances in Optics and Photonics 1 (2009) 1. [8] R. Dorn, S. Quabis, G. Leuchs, Physical Review Letters 91 (2003) [9] Z. Chen, D. Zhao, Optics Letters 37 (2012) [10] R. Peng, B. Yao, S. Yan, W. Zhao, M. Lei, Journal of the Optical Society of America B 26 (2009) [11] Yuichi Koawa, Shunichi Sato, Optics Express 18 (2010) [12] T.A. Nieminen, N.R. Heckenberg, H. Rubinstein-Dunlop, Optics Letters 33 (2008) 122. [13] J. Durnin, Journal of the Optical Society of America A 4 (1987) 651. [14] F. Gori, G. Guattari, C. Padovani, Optics Communications 64 (1987) 491. [15] F.O. Farrbach, P. Simon, A. Rohrbach, Nature Photonics 4 (2010) 780. [16] R.P. MacDonald, S.A. Boothroyd, T. Okamoto, J. Chrostowski, B.A. Syrett, Optics Communications 122 (1996) 169. [17] V. Garces-Chave, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Nature 419 (2002) 145. [18] B. Lu, W. Huang, B. Zhang, Optics Communications 119 (1995) 6. [19] G. Chen, H. Lin, J. Pu, Journal of Optoelectronics Laser 22 (2011) 946. [20] J. Arlt, K. Dholakia, Optics Communications 177 (2000) 297. [21] M. Anguiano-Morales, A. Martine, M.D. Iturbe-Castillo, S. Chave-Cerda, Applied Optics 46 (2007) [22] Y. Lin, W. Seka, J.H. Eberly, H. Huang, D.L. Brown, Applied Optics 31 (1992) [23] Z. Bouchal, J. Wagner, M. Chlup, Optics Communications 151 (1998) 207. [24] S.H. Tao, X. Yuan, Journal of the Optical Society of America A 21 (2004) [25] I.A. Litvin, M.G. McLaren, A. Forbes, Optics Communications 282 (2009) [26] S.A. Collins, Journal of the Optical Society of America 60 (1970) 1168.