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Formation of optical vortices using coherent laser beam arrays
Optics Communications 282 (2009) 1088–1094
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Formation of optical vortices using coherent laser beam arrays Li-Gang Wang a,b,*, Li-Qin Wang a, Shi-Yao Zhu a,b,c a b c
Institute of Optics, Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 14 August 2008 Received in revised form 28 November 2008 Accepted 4 December 2008
PACS: 10.020 10.060 10.010
a b s t r a c t We present a proposal to generate an optical vortex beam by using the coherent-superposition of multibeams in a radially symmetric configuration. In terms of the generalized Huygens–Fresnel diffraction integral, we have derived the general propagation expression for the coherent radial arrays of laser beams. Using the derived formulae, we have analyzed the effects of the beamlet number N, the separation distance q of the beamlets and the topological charge m on the intensity and phase distributions of the resultant beams. Our simulation results show that optical vortices could be efficiently generated due to the coherent-superposition effect of all beamlets, during the propagation process of the coherent radial array of laser beams with the initial well-organized phase distributions through the free space. In the focusing system, the resultant beam near the focusing plane has the strong rotational effect with the phase helicity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Optical vortices, which contain topological wave front dislocations [1], have attracted intensive attentions in many branches of classical and quantum physics [2,3]. The wave front dislocation refers to a continuous line on which the wave phase is undetermined (singular) and its field amplitude vanishes or, in a mathematical form, it is a zero of a complex function with a nonzero phase circulation around it. Owing to this circulation, the optical wave carries the orbital angular momentum [4]. According to the classification introduced by Nye and Berry [1], there are mainly two types of phase singularities: screw wave front dislocation and edge dislocation, although the mixed screw–edge dislocation occurs in most situations. Optical vortices have many important applications, e.g., optical testing [5], nonlinear optics [6], optical tweezers [7,8], high-resolution fluorescence microscopy [9], lithography [10], quantum entanglement [3,11,12], and stellar coronagraph [13,14]. Nowadays, optical vortices can be generated by various methods such as mode conversions [15,16], computer generated holograms [17], spiral phase plates [18,19] and multi-level spiral phase plate [20], optical wedges [21,22], and adaptive helical mirror [23]. Recently, the interference of several plane waves was used to generate the optical vortices [24]. Vyas and Senthilkumaran [25,26] have proposed the modified Michelson interferometer and the modified Mach–Zehnder interferometer for producing * Corresponding author. Address: Department of Physics, Optics Institute, Zhejiang University, Hangzhou 310027, China. Tel.: +86 13858199471. E-mail address: [email protected] (L.-G. Wang). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.12.004
optical vortex arrays. Petrovic´ [27] have found the stable rotating structures in the optical photonic lattices by using the rotating counter propagating incoherent self-trapped vortex beams. Recently, Izdebskaya et al. [28] first theoretically analyzed and experimentally realized the singular beams with very high orbital angular momentum by using an array of vortex beams whose axes lie on the surface of a hyperboloid of revolution. They further investigated the conditions of structural stability of a symmetric array of off-axis singular beams and mainly discussed how the array parameters affect on the orbital angular momentum of the array [29]. In this paper, we pay attention to the formation of optical vortices during the propagation process of a radial array composed of coherent Gaussian beamlets. In the current scheme, we arrange the fundamental Gaussian beams with the initial well-ordered phases in a radially symmetric configuration and all initial beamlets are weakly overlapped. Then we derive the propagation formula of the radial beam array passing through the first-order linear optical systems. It should also be mentioned that in Refs. [30,31], Desyatnikov and Kivshar have already used the same arrangement as the input ansatz of the optical nonlinear Schrödinger equation, and they have introduced the concept of optical soliton clusters and have mainly focused on the physical mechanism for stabilizing multisoliton bound states in a nonlinear medium. However in our paper, we find that the stable optical vortices can be formed both at the far-field region of the free space and at the focusing plane of the lens optical systems, and the formed vortex is mainly due to the coherent-superposition effect among different beamlets as the propagating distance increases. These results can be readily realized in a laser experiment.
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Our investigation is also different from the previous studies [32–37], where the beam arrays in the radial or rectangular symmetric configurations are investigated in phase-locked and nonphase-locked cases. For the phase-locked case it usually refers to the coherent-superposition of all beamlets with the same initial phases, while for the non-phase-locked case it refers to the incoherent addition of all beamlets [33]. However, unlike the previous phase locked radial laser arrays, each beamlet of the coherent radial beam arrays considered here has the different initial phase in a well-ordered distribution. It is found that the optical vortices could be formed during the propagation process of such radially symmetric beam arrays. Our results have useful application in the optical manipulation of micro-sized particles [7,8,38] and atomic trapping and guiding [39,40].
2. A radial laser array and its propagation Consider a radial laser array consisted of N identical fundamental Gaussian beams, which are located symmetrically on a ring with radius q, as illustrated in Fig. 1a. Each beamlet is assumed to be symmetric with a waist width w0 and different initial phase /j, and the displacement coordinates (aj, bj) of each beamlet’s center are (q cos hj, q sin hj), where hj ¼ pð2j 1Þ=N is the azimuth angle of the j th beamlet and q controls the separation distance d among these beamlets. Here, we consider that different beamlets should be weakly overlapped. The initial beam array is located at the input plane of z = 0, and propagates through an axially symmetric optical system. For the jth initial Gaussian beamlet, its field distribution in the rectangular coordinates is defined by
"
# ðx1 aj Þ2 þ ðy1 bj Þ2 Ej ðx1 ; y1 ; z ¼ 0Þ ¼ G0 exp exp½iuj ; w20
Z
1 1 ik exp½ikL0 Ej ðx1 ; y1 ; 0Þ 2pB 1 1 ik exp ½Aðx21 y21 Þ 2ðx1 x2 þ y1 y2 Þ þ Dðx22 þ y22 Þ dx1 dx1 ; 2B
Ej ðx2 ; y2 ; zÞ ¼
ð2Þ where k ¼ 2p=k is the wave number of the incident beamlet with wavelength k. The first-order linear optical system is denoted by A B the ray transfer matrix , and L0 is the axial optical path C D length from the input plane (z = 0) to the output plane (z). On substituting Eq. (1) into Eq. (2) and after tedious integral calculation, we can obtain the expression of each beamlet’s field distribution at the output plane,
" # 2 h i ikAða2j þ bj Þ iZ r exp ikL0 þ iuj exp Ej ðx2 ; y2 ; zÞ ¼ G0 n 2n ikðaj x2 þ bj y2 Þ ikgðx22 þ y22 Þ exp exp ; n 2
ð3Þ
2
where Z r ¼ kw0 =2 is the Rayleigh distance, n ¼ B iAZ r , and g ¼ ðD iCZr Þ=n. Eq. (3) describes a decentered Gaussian beam which retains the Gaussian property unchanged after propagating through an unapertured ABCD system. Here, we consider the radial beam array is combined with the identical fundamental Gaussian beams having different initial constant phases. During the propagation, each beamlet will interfere with each other, which induces the novel properties with the dark-intensity center, the rotation of the resultant beam and helical phase structure. Because all the beamlets are coherent-superposition, the resultant field distribution of the array at the output plane could be expressed as
ð1Þ
where j ¼ 1; 2; . . . ; N denotes the list of the beamlets, the initial phase uj ¼ pmð2j 1Þ=N, m is related to the topological charge of the resulted beam, and G0 is a constant. In principle, we have no requirement on the number m, which can be integer or fractional number. For a fractional number, we may have the resultant optical vortex with a fractional topological charge. In the following analysis, we only discuss the integer cases. The output field distribution of each beamlet passing through a first-order optical system is described by the generalized Huygens–Fresnel diffraction integral [41,42], which can be written as
Z
Eðx2 ; y2 ; zÞ ¼
N X
Ej ðx2 ; y2 ; zÞ
j¼1
iZ r ikAq2 ikgðx22 þ y22 Þ exp½ikL0 exp exp 2 n 2n N X ikðaj x2 þ bj y2 Þ iZ r exp þ iuj ¼ G0 n n j¼1 2 2 2 ikAq ikgðx2 þ y2 Þ exp exp½ikL0 exp 2 2n N X ikqfx2 cos½pð2j 1Þ=N þ y2 sin½pð2j 1Þ=Ng exp n j¼1 pm þi ð2j 1Þ ; ð4Þ N
¼ G0
Using Eq. (4), the corresponding intensity distribution at the output plane reads as
Iðx2 ; y2 ; zÞ ¼ Eðx2 ; y2 ; zÞE ðx2 ; y2 ; zÞ;
ð5Þ
and its phase distribution could be readily obtained from:
u ¼ tan1 fIm½Eðx2 ; y2 ; zÞ=Re½Eðx2 ; y2 ; zÞgu 2 ½0; 2pÞ;
Fig. 1. (a) Schematic of a radial laser arrays with N = 3, 5, 8 identical fundamental Gaussian beams, which are uniformly located on a ring with radius q at the azimuth angle hj j ¼ 1; . . . ; N, and d denotes the separation distance between two neighboring beamlets. (b) Schematic of a focusing lens optical system.
ð6Þ
Eqs. (4)–(6) provide a general description of the radial beam arrays passing through an axially symmetric first-order optical system. In the following discussions, we use Eqs. (4)–(6) to study the propagation property of such a radial array and how the optical vortices produce from the coherent-superposition of the beamlets. The location of a vortex within the intensity distribution could be observed by examining the phase structure as follows: a change from 0 to 2p around a point indicates a screw-type vortex and a phase discontinuous line indicates an edge dislocation. In the following simulations, we take the parameters as follows: k ¼ 632:8 nm, w0 ¼ 1 mm, and consequently Z r 5 m.
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3. Evolution of the radial beam array and formation of optical vortices In this section, we use Eqs. (4)–(6) to graphically illustrate the propagation properties of the radial coherent beam array with each individual Gaussian beam having a different initial constant phase. We will graphically, show how the optical vortex forms during the propagation process of the array. Figs. 2–4 show the typical evolutions of the intensity and resultant phase distributions for the different radial arrays propagating through the free spaces. The ray A B 1 z transfer matrix of the free space is given by ¼ , C D 0 1 where z is the propagation distance. From Figs. 2–4, we find that
each beamlet is completely separated from each other at the initial plane z = 0, and gradually interferes with each other during the propagating process. From the intensity evolutions in Figs. 2–4, it is clearly seen that the regions with the maximal intensities gradually rotate. For example, as shown in Fig. 2, as the propagation distance increases, the interference among these three beamlets leads to the rotation phenomena (see the changes of the dashed triangles in Fig. 2). Note that the phase distributions, in Figs. 2– 4a are not the initial phases but the total combination effect. From Figs. 2–4, it is shown that the phase evolution of the array reveals the physical nature of the intensity evolution. Clearly, there is always a screw-type phase singularity at the center position (0, 0), around which the phase changes from 0 to 2p (obviously, see Figs.
Fig. 2. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 3) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr, and (d) z = 10Zr, with other parameters m = 1 and q = 2w0.
Fig. 3. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 5) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr and (d) z = 10Zr, with other parameters m = 1 and q = 3w0.
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Fig. 4. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 8) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr and (d) z = 10Zr, with other parameters m = 1 and q = 4.5w0. In (d), the dashed circle in the phase diagram denotes the edge-like phase dislocation, which corresponds to a zerointensity circle in the intensity profile.
2–4c and d). It is easy to find that there are other phase singularities, in the phase diagram of Fig. 2d, which also correspond to other zero intensities. Therefore, the resultant beam becomes an annular hollow beam at the far-field region. For the beam array with small N, the initial phase dislocations evolve into the isolated phase singularities (i.e., the so-called screw-type dislocations, see the phase diagrams from Fig. 2a–d), and around each phase singularity the beam intensity is close to zero. For the beam array with larger N, by adjusting the parameter q, the resultant beam may have an isolated dark-hollow beam in the inner part, and by a carefully observation we can find a phase singular circle denoted by the dashed circle in Fig. 4d, which corresponds to the phase closed-edge dislocation for larger N. Therefore, the resultant beam gradually forms an annular beam with a zero-intensity center in the inner region and becomes the shape like a gear wheel at the outer region (see Fig. 4a–d). Meanwhile, the inner annular dark-hollow beam is isolated from the other intensity distribution (see Figs. 4d and 7b and c). Here, it should be pointed out that it is easy to observe the singular phase circle in Fig. 7b and c. Actually, such beams with phase singularities are so-called optical vortices which have the rotation phenomena and have the orbital angular moments [2,4,28,29]. We emphasize that the first inner annular dark-hollow beam contains the most energy of the initial input energy, and the intensity distributions of the optical vortices generated by this method become stable in the far-field region of the free space. In order to study propagation properties of the radially coherent beam arrays in the far-field region, we use a lens transformation system to image the far-field intensity distributions at the focusing plane. As shown in Fig. 1b, the ray transfer matrix for an apertureless lens system between the input and output planes could be given by
A B 1 0 1 s 1 z1 þ f ¼ C D 1=f 1 0 1 0 1 z1 =f f þ ð1 s=f Þz1 ; ¼ 1=f 1 s=f
ð7Þ
where s is the distance between the input plane to the thin lens, f is the focal length, and z1 is the distance from the focusing plane to the output plane. Therefore, the far-field properties can be studied di-
rectly by the intensity and phase distributions at the focusing plane (z1 = 0). In Fig. 5, we observe a typical intensity rotational process before and after the focusing plane z1 = 0. It is clear seen that the resultant beam is clockwise rotating around the propagation axis z and transferring its energy into the inner annular dark-hollow beam before the focusing plane, while after the focusing plane the resultant beam is still clockwise rotating but transferring its energy from the inner annular dark-hollow beam to the outside. Near the focusing plane the most of the energy of the beam array is transferred into the inner dark-hollow vortex beam (see Fig. 5d). Therefore, the generated optical vortex by this coherent radial beam array has orbital angular momentum, and near the focusing plane the resultant vortex beam may be used to manipulate and rotate the small particles [7,8]. In order to obtain the high-quality optical vortex with the darkintensity center on the focusing plane, one has to adjust the parameter q. As an example shown in Fig. 6, it is found that in this case when we choose a suitable value of q ¼ 2w0 , the most light energy is transferred into the inner annular dark-hollow beam with few sidelobes; if q becomes smaller, although there is no sidelobe in the focusing intensity distribution, the inner annular dark-hollow beam has a much large scale; if q becomes larger, although the scale of the inner annular dark-hollow beam has a much smaller size, much light energy is transferred into the outside sidelobes of the intensity profile. Similar effect is existed for the radial arrays with different N. Fig. 7 shows the intensity and phase distributions on the focusing plane for different radially coherent beam arrays. In the phase distributions of Fig. 7a and b, the gray curved arrows denote the screw phases around the phase singularities, and each of these phase singularities corresponds to the zero-intensity distribution (i.e., a vortex core). With the increasing of the number N and under the suitable choice of q, one may find double-annular or multiannular dark-hollow beam in the inner region of the intensity distribution, i.e., the vortex core at center forms the most inner darkhollow vortex beam and the second circular vortex strip forms the second annular dark-hollow vortex beam, and so on (see Figs. 7b and c, and 8a). The dark-intensity pattern between two annular
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Fig. 5. The output intensity distributions of the radial beam array (N = 5) through a lens near the focusing plane: (a) z1 = 0.15Zr, (b) z1 = 0.1Zr, (c) z1 = 0.05Zr and (d) z1 = 0, (e) z1 = 0.1Zr, and (f) z1 = 0.2Zr with other parameters m = 1 and q = 3w0, s = 0.2Zr and f = 0.5Zr.
Fig. 6. The intensity (upper) and phase (below) distributions of the different radial beam arrays with different parameters: (a) q = 1.5w0, (b) q = 2w0, and (c) q = 3w0 at the focusing plane z1 = 0, with other parameters N = 6, m = 1, s = 0.2Zr and f = 0.5Zr.
bright patterns always corresponds to a closed-edge phase dislocation. For example, in Fig. 7b and c, the inner dash-curved arrows correspond to the first inner annular dark-hollow vortex beam and the second dot-curved arrows form the second annular vortex pattern; and between these two annular vortex patterns there is a circle of the closed-edge phase dislocation corresponding to a circular dark pattern. Therefore, it is also possible to obtain the multi-annular hollow vortex beams with several homocentric phase helical structures. Fig. 8 shows the effect of the topological charge m on the focusing properties of the radial beam array. It is clearly seen that the most inner dark-hollow region becomes large for the radial array with a large m. It indicates that one can design the desired output optical vortices with different topological charges by controlling the initial phases on the beamlets in the initial radial array. Under the fixed N and other fixed parameters, with the increasing of m the number for the multi-annular hollow vortex beams with the homocentric phase helical structures becomes smaller and smaller.
4. Summary We have proposed a method to generate optical vortices by using the coherent-superposition of Gaussian beamlets in a radially symmetric configuration. The general propagation formulae for the radially coherent beam arrays passing through the first-order linear optical systems are analytically derived in terms of the generalized Huygens–Fresnel diffraction integral. Based on the derived equations, the propagation properties of the coherent radial arrays are analyzed in detail. From the intensity and phase structures, it is found that the resultant beams become optical vortices with the rotating effects and phase helical structures. Our results are different from the previous studies [30,31], where Desyatnikov and Kivshar mainly showed the optical soliton clusters (i.e. multisoliton bound states) like as ‘‘atoms of light” in a homogeneous nonlinear medium. It should be emphasized that although we only consider the propagation of the radial arrays composed of the fundamental Gaussian beams, it easily generalize this method into
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Fig. 7. The intensity (upper) and phase (below) distributions for the different radial beam arrays with: (a) N = 5 and q = 3w0, (b) N = 8 and q = 4.5w0, and (c) N = 10 and q = 5.5w0, at the focusing plane z1 = 0. Note that in (a) and (b) the curved gray arrows encircle the phase singularities that correspond to the dark-intensity regions (upper). The other parameters are s = 0.2Zr, f = 0.5Zr, and m = 1.
Fig. 8. The intensity (upper) and phase (below) distributions for the different radial beam arrays with different topological charges: (a) m = 1, (b) m = 3, and (c) m = 5 at the focusing plane (z1 = 0), with other parameters N = 25 and q = 16w0.
other complex cases such as the beam arrays consisted of the coherent Laguerre–Gaussian beams (i.e. similar to vortex arrays [28,29]) and the arrays consisted of the coherent truncated higher-order Bessel beams [43]. Finally, we have to mention the previous work proposed by Pas’ko et al. [44] where the transversal optical vortex is formed by the interference fields of two Gaussian beams. Our result may have potential applications in rotating and guiding the small-sized particles [45].
Acknowledgments This work was supported by the National Nature Science Foundation of China (No. 10604047), by Zhejiang Province Scientific Research Foundation (Nos. G20630 and G80611) and by the financial
support from RGC of HK Government (No. NSFC 05-06/01) and the financial support from Zhejiang University. References [1] J.F. Nye, M.V. Berry, Proc. Roy. Soc. Lond. Ser. A 336 (1974) 165. [2] J.F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, Institute of Physics Publishing, Berlin, 1999. [3] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412 (2001) 313. [4] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 6320. [5] P. Senthilkumaran, Appl. Opt. 42 (2003) 6314. [6] K. Dholakia, N.B. Simpson, M.J. Padgett, L. Allen, Phys. Rev. A 54 (1996) R3742. [7] A. Ashkin, Biophys. J. 61 (1992) 569. [8] K.T. Gahaghan, G.A. Swartzlander Jr., Opt. Lett. 21 (1996) 827. [9] V. Westphal, S.W. Hell, Phys. Rev. Lett. 94 (2005) 143903. [10] M.D. Levenson, T.J. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, S.M. Tan, J. Microlithogr. Microfabr. Microsyst. 3 (2004) 293.
1094
L.-G. Wang et al. / Optics Communications 282 (2009) 1088–1094
[11] H.H. Arnaut, G.A. Barbosa, Phys. Rev. Lett. 85 (2000) 286. [12] S. Franke-Arnold, S.M. Barnett, M.J. Padgett, L. Allen, Phys. Rev. A 65 (2002) 033823. [13] G. Foo, D.M. Palacios, G.A. Swartzlander, Opt. Lett. 30 (2005) 3308. [14] J.H. Lee, G. Foo, E.G. Johnson, G.A. Swartzlander Jr., Phys. Rev. Lett. 97 (2006) 053901. [15] E. Abramochkin, V. Volostnikov, Opt. Commun. 83 (1991) 123. [16] M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, J.P. Woerd-man, Opt. Commun. 96 (1993) 123. [17] N.R. Heckenberg, R. McDuff, C.P. Smith, A.G. White, Opt. Lett. 17 (1992) 211. [18] M. Beijersbergen, R.P.C. Coerwinker, M. Kristensen, J.P. Woerdman, Opt. Commun. 112 (1994) 321. [19] C. Rotschild, S. Zommer, Sh. Moed, O. Hershcovitz, S.G. Lipson, Appl. Opt. 43 (2004) 2397. [20] V.V. Kotlyar, A.A. Kovalev, Opt. Lett. 33 (2008) 189. [21] Y. Izdebskaya, V. Shvedov, A. Volyar, Opt. Lett. 30 (2005) 2472. [22] X.C. Yuan, B.P.S. Ahluwalia, H.L. Chen, J. Bu, J. Lin, Appl. Phys. Lett. 91 (2007) 051103. [23] K. O’Holleran, M.J. Padgett, M.R. Dennis, Opt. Exp. 14 (2006) 3039. [24] D.P. Ghai, P. Senthilkumaran, R.S. Sirohi, Appl. Opt. 47 (2008) 1378. [25] S. Vyas, P. Senthilkumaran, Appl. Opt. 46 (2007) 2893. [26] S. Vyas, P. Senthilkumaran, Appl. Opt. 46 (2007) 7862. [27] M.S. Petrovic´, Opt. Exp. 14 (2006) 9415.
[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
Y. Izdebskaya, T. Fadeyeva, V. Shvedov, A. Volyar, Opt. Lett. 31 (2006) 2523. Y. Izdebskaya, V. Shvedov, A. Volyar, J. Opt. Soc. Am. A 25 (2008) 171. A.S. Desyatnikov, Y.S. Kivshar, Phys. Rev. Lett. 88 (2002) 053901. A.S. Desyatnikov, Y.S. Kivshar, J. Opt. B: Quant. Semiclass. Opt. 4 (2002) S58. K.M. Abramski, A.D. Colley, H.J. Baker, D.R. Hall, IEEE J. Quant. Electron. 32 (1996) 340. W.D. Bilida, J.D. Strohschein, H.J.J. Seguin, Gas and chemical laser and applications II, in: R.C. Sze, E.A. Dorko (Eds.), Proc. SPIE, 2987, 1997, p. 13. J.D. Strohschein, H.J.J. Seguin, C.E. Capjack, Appl. Opt. 37 (1998) 1045. B. Lü, H. Ma, J. Opt. Am. A 17 (2000) 2005. V. Eckhouse, A.A. Ishaaya, L. Shimshi, N. Davidson, A.A. Friesem, Opt. Lett. 31 (2006) 350. B. Lü, H. Ma, Opt. Commun. 178 (2000) 395. C.L. Zhao, L.G. Wang, X.H. Lu, Phys. Lett. A 363 (2006) 502. A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. S. Chu, J.E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57 (1986) 314. S.A. Collins, J. Opt. Soc. Am. 60 (1970) 1168. Q. Lin, L. Wang, Opt. Commun. 185 (2000) 263. V.S. Ilchenko, M. Mohageg, A.A. Savchenkov, A.B. Matsko, L. Maleki, Opt. Exp. 15 (2007) 5866. V.A. Pas’ko, M.S. Soskin, M.V. Vasnetsov, Opt. Commun. 198 (2001) 49. L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant, K. Dholakia, Science 292 (2001) 912.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Formation of optical vortices using coherent laser beam arrays Li-Gang Wang a,b,*, Li-Qin Wang a, Shi-Yao Zhu a,b,c a b c
Institute of Optics, Department of Physics, Zhejiang University, Hangzhou 310027, China Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 14 August 2008 Received in revised form 28 November 2008 Accepted 4 December 2008
PACS: 10.020 10.060 10.010
a b s t r a c t We present a proposal to generate an optical vortex beam by using the coherent-superposition of multibeams in a radially symmetric configuration. In terms of the generalized Huygens–Fresnel diffraction integral, we have derived the general propagation expression for the coherent radial arrays of laser beams. Using the derived formulae, we have analyzed the effects of the beamlet number N, the separation distance q of the beamlets and the topological charge m on the intensity and phase distributions of the resultant beams. Our simulation results show that optical vortices could be efficiently generated due to the coherent-superposition effect of all beamlets, during the propagation process of the coherent radial array of laser beams with the initial well-organized phase distributions through the free space. In the focusing system, the resultant beam near the focusing plane has the strong rotational effect with the phase helicity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Optical vortices, which contain topological wave front dislocations [1], have attracted intensive attentions in many branches of classical and quantum physics [2,3]. The wave front dislocation refers to a continuous line on which the wave phase is undetermined (singular) and its field amplitude vanishes or, in a mathematical form, it is a zero of a complex function with a nonzero phase circulation around it. Owing to this circulation, the optical wave carries the orbital angular momentum [4]. According to the classification introduced by Nye and Berry [1], there are mainly two types of phase singularities: screw wave front dislocation and edge dislocation, although the mixed screw–edge dislocation occurs in most situations. Optical vortices have many important applications, e.g., optical testing [5], nonlinear optics [6], optical tweezers [7,8], high-resolution fluorescence microscopy [9], lithography [10], quantum entanglement [3,11,12], and stellar coronagraph [13,14]. Nowadays, optical vortices can be generated by various methods such as mode conversions [15,16], computer generated holograms [17], spiral phase plates [18,19] and multi-level spiral phase plate [20], optical wedges [21,22], and adaptive helical mirror [23]. Recently, the interference of several plane waves was used to generate the optical vortices [24]. Vyas and Senthilkumaran [25,26] have proposed the modified Michelson interferometer and the modified Mach–Zehnder interferometer for producing * Corresponding author. Address: Department of Physics, Optics Institute, Zhejiang University, Hangzhou 310027, China. Tel.: +86 13858199471. E-mail address: [email protected] (L.-G. Wang). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.12.004
optical vortex arrays. Petrovic´ [27] have found the stable rotating structures in the optical photonic lattices by using the rotating counter propagating incoherent self-trapped vortex beams. Recently, Izdebskaya et al. [28] first theoretically analyzed and experimentally realized the singular beams with very high orbital angular momentum by using an array of vortex beams whose axes lie on the surface of a hyperboloid of revolution. They further investigated the conditions of structural stability of a symmetric array of off-axis singular beams and mainly discussed how the array parameters affect on the orbital angular momentum of the array [29]. In this paper, we pay attention to the formation of optical vortices during the propagation process of a radial array composed of coherent Gaussian beamlets. In the current scheme, we arrange the fundamental Gaussian beams with the initial well-ordered phases in a radially symmetric configuration and all initial beamlets are weakly overlapped. Then we derive the propagation formula of the radial beam array passing through the first-order linear optical systems. It should also be mentioned that in Refs. [30,31], Desyatnikov and Kivshar have already used the same arrangement as the input ansatz of the optical nonlinear Schrödinger equation, and they have introduced the concept of optical soliton clusters and have mainly focused on the physical mechanism for stabilizing multisoliton bound states in a nonlinear medium. However in our paper, we find that the stable optical vortices can be formed both at the far-field region of the free space and at the focusing plane of the lens optical systems, and the formed vortex is mainly due to the coherent-superposition effect among different beamlets as the propagating distance increases. These results can be readily realized in a laser experiment.
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Our investigation is also different from the previous studies [32–37], where the beam arrays in the radial or rectangular symmetric configurations are investigated in phase-locked and nonphase-locked cases. For the phase-locked case it usually refers to the coherent-superposition of all beamlets with the same initial phases, while for the non-phase-locked case it refers to the incoherent addition of all beamlets [33]. However, unlike the previous phase locked radial laser arrays, each beamlet of the coherent radial beam arrays considered here has the different initial phase in a well-ordered distribution. It is found that the optical vortices could be formed during the propagation process of such radially symmetric beam arrays. Our results have useful application in the optical manipulation of micro-sized particles [7,8,38] and atomic trapping and guiding [39,40].
2. A radial laser array and its propagation Consider a radial laser array consisted of N identical fundamental Gaussian beams, which are located symmetrically on a ring with radius q, as illustrated in Fig. 1a. Each beamlet is assumed to be symmetric with a waist width w0 and different initial phase /j, and the displacement coordinates (aj, bj) of each beamlet’s center are (q cos hj, q sin hj), where hj ¼ pð2j 1Þ=N is the azimuth angle of the j th beamlet and q controls the separation distance d among these beamlets. Here, we consider that different beamlets should be weakly overlapped. The initial beam array is located at the input plane of z = 0, and propagates through an axially symmetric optical system. For the jth initial Gaussian beamlet, its field distribution in the rectangular coordinates is defined by
"
# ðx1 aj Þ2 þ ðy1 bj Þ2 Ej ðx1 ; y1 ; z ¼ 0Þ ¼ G0 exp exp½iuj ; w20
Z
1 1 ik exp½ikL0 Ej ðx1 ; y1 ; 0Þ 2pB 1 1 ik exp ½Aðx21 y21 Þ 2ðx1 x2 þ y1 y2 Þ þ Dðx22 þ y22 Þ dx1 dx1 ; 2B
Ej ðx2 ; y2 ; zÞ ¼
ð2Þ where k ¼ 2p=k is the wave number of the incident beamlet with wavelength k. The first-order linear optical system is denoted by A B the ray transfer matrix , and L0 is the axial optical path C D length from the input plane (z = 0) to the output plane (z). On substituting Eq. (1) into Eq. (2) and after tedious integral calculation, we can obtain the expression of each beamlet’s field distribution at the output plane,
" # 2 h i ikAða2j þ bj Þ iZ r exp ikL0 þ iuj exp Ej ðx2 ; y2 ; zÞ ¼ G0 n 2n ikðaj x2 þ bj y2 Þ ikgðx22 þ y22 Þ exp exp ; n 2
ð3Þ
2
where Z r ¼ kw0 =2 is the Rayleigh distance, n ¼ B iAZ r , and g ¼ ðD iCZr Þ=n. Eq. (3) describes a decentered Gaussian beam which retains the Gaussian property unchanged after propagating through an unapertured ABCD system. Here, we consider the radial beam array is combined with the identical fundamental Gaussian beams having different initial constant phases. During the propagation, each beamlet will interfere with each other, which induces the novel properties with the dark-intensity center, the rotation of the resultant beam and helical phase structure. Because all the beamlets are coherent-superposition, the resultant field distribution of the array at the output plane could be expressed as
ð1Þ
where j ¼ 1; 2; . . . ; N denotes the list of the beamlets, the initial phase uj ¼ pmð2j 1Þ=N, m is related to the topological charge of the resulted beam, and G0 is a constant. In principle, we have no requirement on the number m, which can be integer or fractional number. For a fractional number, we may have the resultant optical vortex with a fractional topological charge. In the following analysis, we only discuss the integer cases. The output field distribution of each beamlet passing through a first-order optical system is described by the generalized Huygens–Fresnel diffraction integral [41,42], which can be written as
Z
Eðx2 ; y2 ; zÞ ¼
N X
Ej ðx2 ; y2 ; zÞ
j¼1
iZ r ikAq2 ikgðx22 þ y22 Þ exp½ikL0 exp exp 2 n 2n N X ikðaj x2 þ bj y2 Þ iZ r exp þ iuj ¼ G0 n n j¼1 2 2 2 ikAq ikgðx2 þ y2 Þ exp exp½ikL0 exp 2 2n N X ikqfx2 cos½pð2j 1Þ=N þ y2 sin½pð2j 1Þ=Ng exp n j¼1 pm þi ð2j 1Þ ; ð4Þ N
¼ G0
Using Eq. (4), the corresponding intensity distribution at the output plane reads as
Iðx2 ; y2 ; zÞ ¼ Eðx2 ; y2 ; zÞE ðx2 ; y2 ; zÞ;
ð5Þ
and its phase distribution could be readily obtained from:
u ¼ tan1 fIm½Eðx2 ; y2 ; zÞ=Re½Eðx2 ; y2 ; zÞgu 2 ½0; 2pÞ;
Fig. 1. (a) Schematic of a radial laser arrays with N = 3, 5, 8 identical fundamental Gaussian beams, which are uniformly located on a ring with radius q at the azimuth angle hj j ¼ 1; . . . ; N, and d denotes the separation distance between two neighboring beamlets. (b) Schematic of a focusing lens optical system.
ð6Þ
Eqs. (4)–(6) provide a general description of the radial beam arrays passing through an axially symmetric first-order optical system. In the following discussions, we use Eqs. (4)–(6) to study the propagation property of such a radial array and how the optical vortices produce from the coherent-superposition of the beamlets. The location of a vortex within the intensity distribution could be observed by examining the phase structure as follows: a change from 0 to 2p around a point indicates a screw-type vortex and a phase discontinuous line indicates an edge dislocation. In the following simulations, we take the parameters as follows: k ¼ 632:8 nm, w0 ¼ 1 mm, and consequently Z r 5 m.
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3. Evolution of the radial beam array and formation of optical vortices In this section, we use Eqs. (4)–(6) to graphically illustrate the propagation properties of the radial coherent beam array with each individual Gaussian beam having a different initial constant phase. We will graphically, show how the optical vortex forms during the propagation process of the array. Figs. 2–4 show the typical evolutions of the intensity and resultant phase distributions for the different radial arrays propagating through the free spaces. The ray A B 1 z transfer matrix of the free space is given by ¼ , C D 0 1 where z is the propagation distance. From Figs. 2–4, we find that
each beamlet is completely separated from each other at the initial plane z = 0, and gradually interferes with each other during the propagating process. From the intensity evolutions in Figs. 2–4, it is clearly seen that the regions with the maximal intensities gradually rotate. For example, as shown in Fig. 2, as the propagation distance increases, the interference among these three beamlets leads to the rotation phenomena (see the changes of the dashed triangles in Fig. 2). Note that the phase distributions, in Figs. 2– 4a are not the initial phases but the total combination effect. From Figs. 2–4, it is shown that the phase evolution of the array reveals the physical nature of the intensity evolution. Clearly, there is always a screw-type phase singularity at the center position (0, 0), around which the phase changes from 0 to 2p (obviously, see Figs.
Fig. 2. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 3) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr, and (d) z = 10Zr, with other parameters m = 1 and q = 2w0.
Fig. 3. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 5) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr and (d) z = 10Zr, with other parameters m = 1 and q = 3w0.
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Fig. 4. Evolution of the intensity (upper) and phase (below) distributions of the radial beam array (N = 8) at different propagating distances: (a) z = 0, (b) z = 2Zr, (c) z = 5Zr and (d) z = 10Zr, with other parameters m = 1 and q = 4.5w0. In (d), the dashed circle in the phase diagram denotes the edge-like phase dislocation, which corresponds to a zerointensity circle in the intensity profile.
2–4c and d). It is easy to find that there are other phase singularities, in the phase diagram of Fig. 2d, which also correspond to other zero intensities. Therefore, the resultant beam becomes an annular hollow beam at the far-field region. For the beam array with small N, the initial phase dislocations evolve into the isolated phase singularities (i.e., the so-called screw-type dislocations, see the phase diagrams from Fig. 2a–d), and around each phase singularity the beam intensity is close to zero. For the beam array with larger N, by adjusting the parameter q, the resultant beam may have an isolated dark-hollow beam in the inner part, and by a carefully observation we can find a phase singular circle denoted by the dashed circle in Fig. 4d, which corresponds to the phase closed-edge dislocation for larger N. Therefore, the resultant beam gradually forms an annular beam with a zero-intensity center in the inner region and becomes the shape like a gear wheel at the outer region (see Fig. 4a–d). Meanwhile, the inner annular dark-hollow beam is isolated from the other intensity distribution (see Figs. 4d and 7b and c). Here, it should be pointed out that it is easy to observe the singular phase circle in Fig. 7b and c. Actually, such beams with phase singularities are so-called optical vortices which have the rotation phenomena and have the orbital angular moments [2,4,28,29]. We emphasize that the first inner annular dark-hollow beam contains the most energy of the initial input energy, and the intensity distributions of the optical vortices generated by this method become stable in the far-field region of the free space. In order to study propagation properties of the radially coherent beam arrays in the far-field region, we use a lens transformation system to image the far-field intensity distributions at the focusing plane. As shown in Fig. 1b, the ray transfer matrix for an apertureless lens system between the input and output planes could be given by
A B 1 0 1 s 1 z1 þ f ¼ C D 1=f 1 0 1 0 1 z1 =f f þ ð1 s=f Þz1 ; ¼ 1=f 1 s=f
ð7Þ
where s is the distance between the input plane to the thin lens, f is the focal length, and z1 is the distance from the focusing plane to the output plane. Therefore, the far-field properties can be studied di-
rectly by the intensity and phase distributions at the focusing plane (z1 = 0). In Fig. 5, we observe a typical intensity rotational process before and after the focusing plane z1 = 0. It is clear seen that the resultant beam is clockwise rotating around the propagation axis z and transferring its energy into the inner annular dark-hollow beam before the focusing plane, while after the focusing plane the resultant beam is still clockwise rotating but transferring its energy from the inner annular dark-hollow beam to the outside. Near the focusing plane the most of the energy of the beam array is transferred into the inner dark-hollow vortex beam (see Fig. 5d). Therefore, the generated optical vortex by this coherent radial beam array has orbital angular momentum, and near the focusing plane the resultant vortex beam may be used to manipulate and rotate the small particles [7,8]. In order to obtain the high-quality optical vortex with the darkintensity center on the focusing plane, one has to adjust the parameter q. As an example shown in Fig. 6, it is found that in this case when we choose a suitable value of q ¼ 2w0 , the most light energy is transferred into the inner annular dark-hollow beam with few sidelobes; if q becomes smaller, although there is no sidelobe in the focusing intensity distribution, the inner annular dark-hollow beam has a much large scale; if q becomes larger, although the scale of the inner annular dark-hollow beam has a much smaller size, much light energy is transferred into the outside sidelobes of the intensity profile. Similar effect is existed for the radial arrays with different N. Fig. 7 shows the intensity and phase distributions on the focusing plane for different radially coherent beam arrays. In the phase distributions of Fig. 7a and b, the gray curved arrows denote the screw phases around the phase singularities, and each of these phase singularities corresponds to the zero-intensity distribution (i.e., a vortex core). With the increasing of the number N and under the suitable choice of q, one may find double-annular or multiannular dark-hollow beam in the inner region of the intensity distribution, i.e., the vortex core at center forms the most inner darkhollow vortex beam and the second circular vortex strip forms the second annular dark-hollow vortex beam, and so on (see Figs. 7b and c, and 8a). The dark-intensity pattern between two annular
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Fig. 5. The output intensity distributions of the radial beam array (N = 5) through a lens near the focusing plane: (a) z1 = 0.15Zr, (b) z1 = 0.1Zr, (c) z1 = 0.05Zr and (d) z1 = 0, (e) z1 = 0.1Zr, and (f) z1 = 0.2Zr with other parameters m = 1 and q = 3w0, s = 0.2Zr and f = 0.5Zr.
Fig. 6. The intensity (upper) and phase (below) distributions of the different radial beam arrays with different parameters: (a) q = 1.5w0, (b) q = 2w0, and (c) q = 3w0 at the focusing plane z1 = 0, with other parameters N = 6, m = 1, s = 0.2Zr and f = 0.5Zr.
bright patterns always corresponds to a closed-edge phase dislocation. For example, in Fig. 7b and c, the inner dash-curved arrows correspond to the first inner annular dark-hollow vortex beam and the second dot-curved arrows form the second annular vortex pattern; and between these two annular vortex patterns there is a circle of the closed-edge phase dislocation corresponding to a circular dark pattern. Therefore, it is also possible to obtain the multi-annular hollow vortex beams with several homocentric phase helical structures. Fig. 8 shows the effect of the topological charge m on the focusing properties of the radial beam array. It is clearly seen that the most inner dark-hollow region becomes large for the radial array with a large m. It indicates that one can design the desired output optical vortices with different topological charges by controlling the initial phases on the beamlets in the initial radial array. Under the fixed N and other fixed parameters, with the increasing of m the number for the multi-annular hollow vortex beams with the homocentric phase helical structures becomes smaller and smaller.
4. Summary We have proposed a method to generate optical vortices by using the coherent-superposition of Gaussian beamlets in a radially symmetric configuration. The general propagation formulae for the radially coherent beam arrays passing through the first-order linear optical systems are analytically derived in terms of the generalized Huygens–Fresnel diffraction integral. Based on the derived equations, the propagation properties of the coherent radial arrays are analyzed in detail. From the intensity and phase structures, it is found that the resultant beams become optical vortices with the rotating effects and phase helical structures. Our results are different from the previous studies [30,31], where Desyatnikov and Kivshar mainly showed the optical soliton clusters (i.e. multisoliton bound states) like as ‘‘atoms of light” in a homogeneous nonlinear medium. It should be emphasized that although we only consider the propagation of the radial arrays composed of the fundamental Gaussian beams, it easily generalize this method into
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Fig. 7. The intensity (upper) and phase (below) distributions for the different radial beam arrays with: (a) N = 5 and q = 3w0, (b) N = 8 and q = 4.5w0, and (c) N = 10 and q = 5.5w0, at the focusing plane z1 = 0. Note that in (a) and (b) the curved gray arrows encircle the phase singularities that correspond to the dark-intensity regions (upper). The other parameters are s = 0.2Zr, f = 0.5Zr, and m = 1.
Fig. 8. The intensity (upper) and phase (below) distributions for the different radial beam arrays with different topological charges: (a) m = 1, (b) m = 3, and (c) m = 5 at the focusing plane (z1 = 0), with other parameters N = 25 and q = 16w0.
other complex cases such as the beam arrays consisted of the coherent Laguerre–Gaussian beams (i.e. similar to vortex arrays [28,29]) and the arrays consisted of the coherent truncated higher-order Bessel beams [43]. Finally, we have to mention the previous work proposed by Pas’ko et al. [44] where the transversal optical vortex is formed by the interference fields of two Gaussian beams. Our result may have potential applications in rotating and guiding the small-sized particles [45].
Acknowledgments This work was supported by the National Nature Science Foundation of China (No. 10604047), by Zhejiang Province Scientific Research Foundation (Nos. G20630 and G80611) and by the financial
support from RGC of HK Government (No. NSFC 05-06/01) and the financial support from Zhejiang University. References [1] J.F. Nye, M.V. Berry, Proc. Roy. Soc. Lond. Ser. A 336 (1974) 165. [2] J.F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, Institute of Physics Publishing, Berlin, 1999. [3] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412 (2001) 313. [4] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 6320. [5] P. Senthilkumaran, Appl. Opt. 42 (2003) 6314. [6] K. Dholakia, N.B. Simpson, M.J. Padgett, L. Allen, Phys. Rev. A 54 (1996) R3742. [7] A. Ashkin, Biophys. J. 61 (1992) 569. [8] K.T. Gahaghan, G.A. Swartzlander Jr., Opt. Lett. 21 (1996) 827. [9] V. Westphal, S.W. Hell, Phys. Rev. Lett. 94 (2005) 143903. [10] M.D. Levenson, T.J. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, S.M. Tan, J. Microlithogr. Microfabr. Microsyst. 3 (2004) 293.
1094
L.-G. Wang et al. / Optics Communications 282 (2009) 1088–1094
[11] H.H. Arnaut, G.A. Barbosa, Phys. Rev. Lett. 85 (2000) 286. [12] S. Franke-Arnold, S.M. Barnett, M.J. Padgett, L. Allen, Phys. Rev. A 65 (2002) 033823. [13] G. Foo, D.M. Palacios, G.A. Swartzlander, Opt. Lett. 30 (2005) 3308. [14] J.H. Lee, G. Foo, E.G. Johnson, G.A. Swartzlander Jr., Phys. Rev. Lett. 97 (2006) 053901. [15] E. Abramochkin, V. Volostnikov, Opt. Commun. 83 (1991) 123. [16] M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, J.P. Woerd-man, Opt. Commun. 96 (1993) 123. [17] N.R. Heckenberg, R. McDuff, C.P. Smith, A.G. White, Opt. Lett. 17 (1992) 211. [18] M. Beijersbergen, R.P.C. Coerwinker, M. Kristensen, J.P. Woerdman, Opt. Commun. 112 (1994) 321. [19] C. Rotschild, S. Zommer, Sh. Moed, O. Hershcovitz, S.G. Lipson, Appl. Opt. 43 (2004) 2397. [20] V.V. Kotlyar, A.A. Kovalev, Opt. Lett. 33 (2008) 189. [21] Y. Izdebskaya, V. Shvedov, A. Volyar, Opt. Lett. 30 (2005) 2472. [22] X.C. Yuan, B.P.S. Ahluwalia, H.L. Chen, J. Bu, J. Lin, Appl. Phys. Lett. 91 (2007) 051103. [23] K. O’Holleran, M.J. Padgett, M.R. Dennis, Opt. Exp. 14 (2006) 3039. [24] D.P. Ghai, P. Senthilkumaran, R.S. Sirohi, Appl. Opt. 47 (2008) 1378. [25] S. Vyas, P. Senthilkumaran, Appl. Opt. 46 (2007) 2893. [26] S. Vyas, P. Senthilkumaran, Appl. Opt. 46 (2007) 7862. [27] M.S. Petrovic´, Opt. Exp. 14 (2006) 9415.
[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
Y. Izdebskaya, T. Fadeyeva, V. Shvedov, A. Volyar, Opt. Lett. 31 (2006) 2523. Y. Izdebskaya, V. Shvedov, A. Volyar, J. Opt. Soc. Am. A 25 (2008) 171. A.S. Desyatnikov, Y.S. Kivshar, Phys. Rev. Lett. 88 (2002) 053901. A.S. Desyatnikov, Y.S. Kivshar, J. Opt. B: Quant. Semiclass. Opt. 4 (2002) S58. K.M. Abramski, A.D. Colley, H.J. Baker, D.R. Hall, IEEE J. Quant. Electron. 32 (1996) 340. W.D. Bilida, J.D. Strohschein, H.J.J. Seguin, Gas and chemical laser and applications II, in: R.C. Sze, E.A. Dorko (Eds.), Proc. SPIE, 2987, 1997, p. 13. J.D. Strohschein, H.J.J. Seguin, C.E. Capjack, Appl. Opt. 37 (1998) 1045. B. Lü, H. Ma, J. Opt. Am. A 17 (2000) 2005. V. Eckhouse, A.A. Ishaaya, L. Shimshi, N. Davidson, A.A. Friesem, Opt. Lett. 31 (2006) 350. B. Lü, H. Ma, Opt. Commun. 178 (2000) 395. C.L. Zhao, L.G. Wang, X.H. Lu, Phys. Lett. A 363 (2006) 502. A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. S. Chu, J.E. Bjorkholm, A. Ashkin, A. Cable, Phys. Rev. Lett. 57 (1986) 314. S.A. Collins, J. Opt. Soc. Am. 60 (1970) 1168. Q. Lin, L. Wang, Opt. Commun. 185 (2000) 263. V.S. Ilchenko, M. Mohageg, A.A. Savchenkov, A.B. Matsko, L. Maleki, Opt. Exp. 15 (2007) 5866. V.A. Pas’ko, M.S. Soskin, M.V. Vasnetsov, Opt. Commun. 198 (2001) 49. L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant, K. Dholakia, Science 292 (2001) 912.