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Transversal optical vortex
15 October 2001
Optics Communications 198 (2001) 49±56
www.elsevier.com/locate/optcom
Transversal optical vortex V.A. Pas'ko, M.S. Soskin, M.V. Vasnetsov * Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev 28, Ukraine Received 14 May 2001; accepted 1 August 2001
Abstract A structure of wavefront edge dislocations and associated ``transversal'' optical vortices in an interference ®eld of two two-dimensional Gaussian beams is analyzed. It was shown that the optical vortex rotation is directed toward the area of higher phase velocity in the interference ®eld (the origin of the phase velocity variation is due to the Gouy eect). The conditions for the reversal of the sign of rotation were found as well as for annihilation of two edge dislocations. Topological reaction of ``unfolding'' of an edge dislocation, which happens when vortex collides with a phase saddle, is studied in details. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.25 Keywords: Optical vortex; Phase singularity; Wavefront dislocation
1. Introduction It was shown in last decades that light ¯ux can produce vortices, similar to vortices in a liquid [1]. The vortex appears around a wavefront dislocation, which is a continuous line in space, where the ®eld amplitude vanishes and the phase is undetermined (singular). According to the classi®cation introduced by Nye and Berry [2], a monochromatic light wave can possess two main types of phase singularities: screw wavefront dislocation and edge dislocation, while mixed edge-screw dislocation is most common situation. Nowadays the term ``optical vortex'' (OV) introduced in Ref. [3] became widely used, re¯ecting the general feature
* Corresponding author. Tel.: +380-44-265-14-22; fax: +38044-265-15-89. E-mail address: [email protected] (M.V. Vasnetsov).
of phase singularities: phase circulation around the dislocation line. Therefore pure screw dislocation is a core of a ``longitudinal'' OV, and edge dislocation produces a ``transversal'' OV, with respect to the wave propagation direction. In our recent paper [4] we have analyzed how an edge dislocation of a wavefront can be created in an interference ®eld of two paraxial Gaussian beams. Destructive interference of co-axial beams results in appearance of a zero-amplitude circle, whose radius and position are determined by relative phases and amplitudes of the beams. Around the circle, which is a circular edge dislocation, a sub-wavelength area of circular light ¯ow was detected within a loop of a separatrix, which divides the light current within and outside the dislocation. The separatrix possesses a self-crossing in a point of the light current stagnation (phase saddle) which, depending on the ratio of the interfering beams amplitudes, can be located as
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 8 7 - 0
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
within, as outside the zero-amplitude circle. Therefore, two possible directions of the torous ¯ow within the separatrix loop can be realized. There is also a situation when the saddle collides with the vortex, what results in the reversing the sign of light circulation and a birth of two new circular edge dislocations. The purpose of the present paper is to describe in more details the behavior of a transversal OV and describe a topological reaction, which occurs in the saddle±vortex collision. The central question is to ®nd the condition, which determines the direction of a transversal OV circulation around the dislocation line. In contrast to the con®guration used in Ref. [4], we now simplify the problem to two-dimensional (2D) (cylindrical) waves. Actually, this approach gives the same results as for three-dimensional (3D) Gaussian beams, because the reactions are governed by general topological laws. However, the choice of 2D waves for the ®eld description makes possible creation of linear edge dislocations and does not require any account for polarization components but one, what is important for the Poynting vector calculation. Also, there is a possibility to compare the results with a general constrains derived in Ref. [5].
2. Interference of 2D Gaussian beams The geometry of waves is explained in Fig. 1. A structure of a single 2D Gaussian beam is shown in Fig. 1(a), with the beam waist at z 0 position. The beam is polarized along Y-axis and propagates along Z-axis, with diraction spread out in X-axis directions. For this 2D beam of Gaussian shape the solution satisfying paraxial wave equation looks as follows: r w0 x2 kx2 E x; z EG exp i exp w w2 2R z 1 z arctan kz U ; 1 2 zR where EG is the amplitude parameter, w0 is the waist parameter, k is the wave number, zR is the Rayleigh range, zR kw20 =2, R z is the radius of
Fig. 1. (a) A schematic view of a 2D Gaussian beam with the waist located at z 0 position. (b) Destructive interference of two 2D Gaussian beams with dierent waist parameters. Solid lines are Gaussian amplitude distributions in the waist plane. The resulting amplitude distribution in the waist is shown by dashed line. Negative amplitude denotes the p phase shift. (c) A wavefront sequence near the waist (U1 1): there is a lack of one wavefront between the dislocation lines. The wavefront ledge before the waist is directed along the direction of the wave propagation and opposite after the waist.
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the wavefront curvature in the XZ plane, 1=2 w w0 1 z2 =z2R , U is a constant phase term. The solution (1) is very similar to the description of 3D Gaussian beam, except another degree of amplitude diminishing with the propagation path z, w0 =w1=2 instead of (w0 =w) for 3D Gaussian beam, and coecient 1=2 before the Gouy phase shift arctan(z=zR ). To produce a pair of edge dislocations parallel to the Y-axis in the plane z 0, let us make a superposition of two 2D Gaussian beams with different amplitudes E1 > E2 , waist parameters w2 > w1 and a phase shift p between their phases U1 and U2 very similar to how it has been done for 3D Gaussian beams [4]: x2 E x; z 0 E1 exp exp iU1 w2 1 2 x E2 exp 2 exp iU2 : w22 At the waist plane, both waves have plane wavefront, and their relative phases are locked as U1 U2 p, thus producing destructive interference, as shown in Fig. 1(b). The edge dislocation position (zero amplitude of the ®eld) is determined by s w21 w22 x0 ln E1 =E2 : 3 w22 w21 For the analysis below, we de®ne the ratio E2 =E1 as the governing parameter g. To ®nd a wavefront structure nearby the waist, and, generally, how the phase behaves at the vicinity of the dark line, we have to calculate the interference ®eld as a sum of two complex amplitudes E x; z E1 x; z E2 x; z and then calculate the resulting phase U x; z: tan U x; z
Im E x; z : Re E x; z
4
Thus the equation for a wavefront family U x; z 0; 2p; 4p; . . . can be found as ImE x; z 0. A family of wavefronts (crests) is shown in Fig. 1(c) for g 0:83, w1 10, w2 100, U1 1 (here and below all ranges are normalized for k 1). First, it is seen from Fig. 1(c) that there is a lack of one wavefront inside the interval bordered by edge dislocations. Before the waist, the wavefront
51
has a ledge along the direction of propagation, after the waist the ledge is directed opposite. The ledge is smoothed far from the waist and the resulting lack of the wavefront results in a small wavelength dierence within the interval between the dislocations and outside, or phase velocity variation. The phase velocity is larger within the interval between the dislocations, the physical reason is the Gouy eect [6]: a spatially compressed light beam increases the phase velocity on the axis. To calculate the phase velocity of a 2D Gaussian beam in the waist, Eq. (1) can be applied x2 vc 1 2 2zR
1 2kzR
1
;
5
where c is the speed ofplight. There is a distance xv where v c: xv w0 = 2. When x < xv , the phase velocity exceeds c, and vice versa. Due to this phase velocity variation, wavefront being plane at the waist gradually transforms to cylindrical at the far ®eld. We can also calculate the phase velocity for the superposition of the beams vR : vR
E1 x E2 xv1 v2 ; v2 E1 x v1 E2 x
6
where vR , v1 and v2 are functions of the transverse coordinate x. For the case of constructive interference, minus sign in the numerator and denominator will be replaced by plus. The higher transversal compression of a beam results in the higher phase velocity, therefore a narrow beam with the waist parameter w1 is a little bit faster at the waist than the wide beam with the waist parameter w2 . We note, however, that the possible presence of an edge dislocation can drastically perturb the smooth velocity dependence along the transversal coordinate nearby the dislocation. Fig. 2 shows a family of equiphase lines in the cross-section XZ in the vicinity of the edge dislocation, for g 0:83. There is an additional wavefront outside the inner stripe between the dislocation lines, and a point where it enters the periodic sequence: here, at the phase saddle, lines of U x; z 0 and U x; z 2p join.
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the zero-amplitude line is localized in a very narrow region. In comparison with the wavelength, the separation between the vortex and the saddle is extremely small and amounts of about 10 2 of the wavelength. 3. Topological reactions in an interference ®eld of 2D Gaussian beams
Fig. 2. Lines of equal phase of the interference ®eld produced by two 2D Gaussian beams in the cross-section XZ nearby the edge dislocation, for g 0:83, U1 0. Saddle point is indicated as a circle, vortex as a dot. There is a p-hop of the phase at the zero-amplitude point (core of the vortex).
The equation for determination of the saddle position xs can be obtained as the coordinate of the point where a wavefront U x; z 0 splits kx2 1 kx2 1 k 2 g k 2 2zR2 2zR1 2zR1 2z R2 2 x x2 exp : 7 w21 w22 Analysis shows that the position of the saddle can be as inside the interval x0 (xs < x0 ), as outside it (xs > x0 ). First, we shall analyze the case xs < x0 which is similar to the described in Ref. [4] circular edge dislocation structure. The phase map looks identical, manifesting the vortices around zeroamplitude lines for g 0:83 (Fig. 2), and we can seek for some particular details now. The distance between vortex and saddle Dx depends on the beam parameters and can be estimated from Eq. (7) for w2 w1 as: Dx
2x20 w21 : 2k 2 w21 x0
8
The physical sense of the saddle point is the condition of the energy current stagnation: vortex ¯ow is exactly compensated by the longitudinal ¯ow, and a line of standing wave is created. Therefore the vorticity area is ended in the saddle point, and the whole torous light current around
With increasing g, x0 diminishes according to Eq. (3), and two dislocation lines come closer as well as saddles. The saddles, which are located inside the interval between the vortices, will meet at x 0 point when g zR2 2kzR1 1= zR1 2kzR2 1. This is the ®rst stage of the collapse (Fig. 3(a)). Then saddles appear again separately, but are located now symmetrical on the Z-axis (Fig. 3(b)). The maximum distance between the saddles will be reached for g 0:995 and amounts approximately to 1. In this moment the distance between vortices becomes equal to the distance between saddles. With further increase of g the saddles as well as vortices come closer. Finally, vortices and saddles collide and annihilate altogether at the point x 0 (Fig. 3(c)). With decreasing of g, saddle point moves toward the vortex and meets it when g g0 exp w21 w22 =2 w21 w22 (this condition corresponds to the coincidence of the dislocation line to the line of equality of phase velocities of the interfering waves, as will be discussed below). In the moment of this collision a saddle±vortex point appears, which is a ``pathological'' topological object. After the crossing event, the saddle appears outside the dislocation line, and thus the light ¯ow circulation around the dislocation reverses the direction. This reaction is accompanied with two similar objects creation outside the plane z 0, as Fig. 4 shows. These two new dislocations have the same ``sign'', or direction of the current circulation around them, as the initial one, so there is again a lack of one wavefront in the central part of the beam. If we start from smaller value of g, g < g0 which corresponds to the existence of three edge dislocations, and then gradually increase g until it reaches g0 value, the topological reaction will look as follows: three parallel zero-amplitude lines
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
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Fig. 4. Lines of equal phase in the cross-section XZ. Both wave crests (solid lines) and troughs (dashed lines) are shown, as calculated from the condition ImE x; z 0, U1 0:1. Position of vortices is shown by dots, saddles are not shown here.
the total number of critical points in the XZ crosssection is six. The signs of vortices, which are out of the waist plane (``positive''), are opposite to the sign of vortex localized in the waist plane (``negative''). After the collision, the remaining objects are one vortex with positive sign and one saddle. Of course, we discuss the events happening only in the x > 0 half plane, in the x < 0 half plane the same process happens symmetrically. Let us examine now how new edge dislocations can sprout from the saddle±vortex line. First, we have to ®nd the conditions for edge dislocation to exist out of the waist plane. There are two necessary conditions, one for destructive interference, another for the equality of the interfering waves amplitudes. The phase condition looks as follows: kx2 2R1 z
Fig. 3. Collapse of edge dislocations. (a) Equiphase lines around the dislocations which are close to the beam center, saddle points have joint. (b) Vortices approach closer, saddles are split along the Z-axis. (c) All topological objects collide and annihilate at g 1.
accompanied with three lines of stagnation (phase saddle lines) come closer until the junction. Altogether, there are three vortices and three saddles,
1 z arctan 2 zR1
kx2 1 z arctan 0: zR2 2R2 z 2 9
There is a trivial solution z 0 (the waist plane), where two initial edge dislocations are located, and a surface of destructive interference where new edge dislocations can appear 1 z2 z2R1 z2 z2R2 2 x kz z2R2 z2R1 z z arctan arctan ; 10 zR1 zR2
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
natural, however, that the axial velocity is usually larger: diraction is responsible for it. Ordinary focusing cannot change this feature, and some tricks are necessary to apply for the construction of the wanted beam. As it is seen from formula (6), subtraction of a ``slow'' beam from a ``fast'' one leads to the increase of its axial velocity, and coherent addition will decrease it. Therefore we have chosen 2D analog of a Laguerre±Gaussian mode LG01 as the fast beam Fig. 5. Solid line: cross-section of the surface of destructive interference (p phase dierence between interfering beams), for x > 0. Dashed lines: cross-sections of the surfaces of the amplitude equality. For g > g0 there is only one possible crossing point and therefore one zero-amplitude line (edge dislocation). For g < g0 there are three points of the crossing and therefore three edge dislocations (saddle points are not shown here).
which can be approximated nearby the waist as x2
zR1 zR2 2z2 2z2R2 zR1 zR2 z2 R1 : k zR1 zR2 3kzR1 zR2 zR1 zR2
11
The cross-section of the surface of the destructive interference is shown in Fig. 5 by solid line. To ®nd the location of edge dislocation out of the waist plane, the surface of the amplitude equality can be found also, as shown in Fig. 5 by dashed line. Its position depends on the governing parameter g, whereas the line of the destructive interference does not depend on the amplitude ratio of the beams. This line crosses the X-axis exactly in the point which corresponds to the equality of phase velocities of the interfering waves. This coincidence is hardly occasional, and can give a hint for understanding of the vortex behavior. The consequence can be made: the vortex rotation is directed toward the area of higher phase velocity.
4. Inversion of an optical vortex circulation To prove this assumption, we have to construct somehow a beam, which possesses a pair of edge dislocations at the waist and the phase velocity between them smaller than peripheral one. It is
ELG E1
r 2 w0 4x 1 exp w w2 5 z i arctan ikz ; 2 zR
x2 kx2 i 2 w 2R z 12
and 2D Gaussian beam with the amplitude parameter E2 and the same waist parameter w0 as the slow beam. Again, the amplitude ratio g E2 =E1 will serve as the governing parameter. The 2D LG01 mode possesses increased Gouy phase shift, what insures higher phase velocity, and inherent phase defect in a form of a p step along the surfaces x2 w20 1 z2 =z2R , where amplitude ELG gets zero value. A coherent in-phase (or out of phase) addition of a 2D Gaussian beam will destroy the zero-amplitude surfaces, leaving the only two zeroamplitude lines at the waist plane, which are linear edge dislocations. Fig. 6(a) shows the result of the destructive interference of the beams. As the amplitude of the 2D LG01 mode in the waist is negative within the interval 2w0 , the subtraction of the 2D Gaussian beam will diminish the axial phase velocity here, whereas outside the interval the interference will increase the phase velocity, what is necessary for our task. The resulting wavefront family is shown in Fig. 6(b). In contrast to Fig. 1(c), an additional wavefront sheet appears within the interval bordered by edge dislocations. The detailed structure of the phase around the dislocation is shown in Fig. 7. The saddle position now is outside the interval between dislocations, and the light vortex rotates around dislocations from interior to outside the interval. Moreover, we can add the slow 2D Gaussian beam instead of the subtraction, and obtain the reversed situation.
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
55
Fig. 7. Lines of equal phase of the destructive interference ®eld produced by 2D LG01 mode and 2D Gaussian beam in the crosssection XZ nearby the edge dislocation, for g 0:5 (compare with Fig. 2).
5. Conclusion
Fig. 6. The same as Fig. 1(b) and (c), but for the destructive interference ®eld produced by 2D LG01 mode and 2D Gaussian beam. (a) Amplitude distribution in the waist, dashed line is the resulting amplitude (negative amplitude denotes the p phase shift). (b) A wavefront sequence near the waist: there is an extra wavefront between the edge dislocations. The ledge is directed backward to the direction of the wave propagation before the waist and along the direction of propagation after the waist.
This veri®cation absolutely agrees with the expectations. A single edge dislocation can be constructed in a similar way, if we take a 2D mode r w0 x x2 kx2 exp E x; z E1 i ww w2 2R z 3 z i arctan ikz 13 2 zR as the fast beam and add/subtract a 2D Gaussian beam.
Summarizing, we have found how the transversal OV, created in an interference ®eld of two 2D Gaussian beams, transforms in topological reactions of edge dislocation collapse and ``unfolding'' [7]. The results obtained are valid as well for 3D beams. The rule for determination of the direction of a transversal OV circulation around an edge dislocation was established: the light vortex rotates toward the region of the higher phase velocity. This rule can be used for the explanation of the natural structure of phase circulation around dark rings (circular edge dislocations) observed by Karman et al. [8]. The physical reason of this feature is not clear enough and this fundamental property of light propagation seems worthwhile for further investigations.
Acknowledgements This work was supported by EOARD, Air Force Oce of Scienti®c Research and Development, Air Force Research Lab, USA (Partner Project P051).
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References [1] M. Vasnetsov, K. Staliunas (Eds.), Optical Vortices, Nova Science Publishers, New York, 1999. [2] J.F. Nye, M.V. Berry, Proc. Roy. Soc. Lond. A 336 (1974) 165. [3] P. Cullet, L. Gil, F. Rocca, Opt. Commun. 73 (1989) 403. [4] M.V. Vasnetsov, V.N. Gorshkov, I.G. Marienko, M.S. Soskin, Opt. Spectrosc. 88 (2000) 260 [Optika Spectrosk. 88 (2000) 298 (in Russian)].
[5] J.F. Nye, J.V. Hajnal, J.H. Hannay, Proc. Roy. Soc. Lond. A 417 (1988) 7. [6] M. Gouy, Compt. Rendue Acad. Sci. Paris 110 (1890) 1251; A.E. Siegman, Lasers, University Science Book, Mill Valley, 1986, p. 1251, pp. 682±683. [7] J.F. Nye, J. Opt. Soc. Am. A 15 (1998) 1132. [8] G.P. Karman, M.W. Beijersbergen, A. van Duijl, J.P. Woerdman, Opt. Lett. 22 (1997) 1503; G.P. Karman, J.P. Woerdman, J. Opt. Soc. Am. A 15 (1998) 2862.
Optics Communications 198 (2001) 49±56
www.elsevier.com/locate/optcom
Transversal optical vortex V.A. Pas'ko, M.S. Soskin, M.V. Vasnetsov * Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev 28, Ukraine Received 14 May 2001; accepted 1 August 2001
Abstract A structure of wavefront edge dislocations and associated ``transversal'' optical vortices in an interference ®eld of two two-dimensional Gaussian beams is analyzed. It was shown that the optical vortex rotation is directed toward the area of higher phase velocity in the interference ®eld (the origin of the phase velocity variation is due to the Gouy eect). The conditions for the reversal of the sign of rotation were found as well as for annihilation of two edge dislocations. Topological reaction of ``unfolding'' of an edge dislocation, which happens when vortex collides with a phase saddle, is studied in details. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.25 Keywords: Optical vortex; Phase singularity; Wavefront dislocation
1. Introduction It was shown in last decades that light ¯ux can produce vortices, similar to vortices in a liquid [1]. The vortex appears around a wavefront dislocation, which is a continuous line in space, where the ®eld amplitude vanishes and the phase is undetermined (singular). According to the classi®cation introduced by Nye and Berry [2], a monochromatic light wave can possess two main types of phase singularities: screw wavefront dislocation and edge dislocation, while mixed edge-screw dislocation is most common situation. Nowadays the term ``optical vortex'' (OV) introduced in Ref. [3] became widely used, re¯ecting the general feature
* Corresponding author. Tel.: +380-44-265-14-22; fax: +38044-265-15-89. E-mail address: [email protected] (M.V. Vasnetsov).
of phase singularities: phase circulation around the dislocation line. Therefore pure screw dislocation is a core of a ``longitudinal'' OV, and edge dislocation produces a ``transversal'' OV, with respect to the wave propagation direction. In our recent paper [4] we have analyzed how an edge dislocation of a wavefront can be created in an interference ®eld of two paraxial Gaussian beams. Destructive interference of co-axial beams results in appearance of a zero-amplitude circle, whose radius and position are determined by relative phases and amplitudes of the beams. Around the circle, which is a circular edge dislocation, a sub-wavelength area of circular light ¯ow was detected within a loop of a separatrix, which divides the light current within and outside the dislocation. The separatrix possesses a self-crossing in a point of the light current stagnation (phase saddle) which, depending on the ratio of the interfering beams amplitudes, can be located as
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 8 7 - 0
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
within, as outside the zero-amplitude circle. Therefore, two possible directions of the torous ¯ow within the separatrix loop can be realized. There is also a situation when the saddle collides with the vortex, what results in the reversing the sign of light circulation and a birth of two new circular edge dislocations. The purpose of the present paper is to describe in more details the behavior of a transversal OV and describe a topological reaction, which occurs in the saddle±vortex collision. The central question is to ®nd the condition, which determines the direction of a transversal OV circulation around the dislocation line. In contrast to the con®guration used in Ref. [4], we now simplify the problem to two-dimensional (2D) (cylindrical) waves. Actually, this approach gives the same results as for three-dimensional (3D) Gaussian beams, because the reactions are governed by general topological laws. However, the choice of 2D waves for the ®eld description makes possible creation of linear edge dislocations and does not require any account for polarization components but one, what is important for the Poynting vector calculation. Also, there is a possibility to compare the results with a general constrains derived in Ref. [5].
2. Interference of 2D Gaussian beams The geometry of waves is explained in Fig. 1. A structure of a single 2D Gaussian beam is shown in Fig. 1(a), with the beam waist at z 0 position. The beam is polarized along Y-axis and propagates along Z-axis, with diraction spread out in X-axis directions. For this 2D beam of Gaussian shape the solution satisfying paraxial wave equation looks as follows: r w0 x2 kx2 E x; z EG exp i exp w w2 2R z 1 z arctan kz U ; 1 2 zR where EG is the amplitude parameter, w0 is the waist parameter, k is the wave number, zR is the Rayleigh range, zR kw20 =2, R z is the radius of
Fig. 1. (a) A schematic view of a 2D Gaussian beam with the waist located at z 0 position. (b) Destructive interference of two 2D Gaussian beams with dierent waist parameters. Solid lines are Gaussian amplitude distributions in the waist plane. The resulting amplitude distribution in the waist is shown by dashed line. Negative amplitude denotes the p phase shift. (c) A wavefront sequence near the waist (U1 1): there is a lack of one wavefront between the dislocation lines. The wavefront ledge before the waist is directed along the direction of the wave propagation and opposite after the waist.
V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the wavefront curvature in the XZ plane, 1=2 w w0 1 z2 =z2R , U is a constant phase term. The solution (1) is very similar to the description of 3D Gaussian beam, except another degree of amplitude diminishing with the propagation path z, w0 =w1=2 instead of (w0 =w) for 3D Gaussian beam, and coecient 1=2 before the Gouy phase shift arctan(z=zR ). To produce a pair of edge dislocations parallel to the Y-axis in the plane z 0, let us make a superposition of two 2D Gaussian beams with different amplitudes E1 > E2 , waist parameters w2 > w1 and a phase shift p between their phases U1 and U2 very similar to how it has been done for 3D Gaussian beams [4]: x2 E x; z 0 E1 exp exp iU1 w2 1 2 x E2 exp 2 exp iU2 : w22 At the waist plane, both waves have plane wavefront, and their relative phases are locked as U1 U2 p, thus producing destructive interference, as shown in Fig. 1(b). The edge dislocation position (zero amplitude of the ®eld) is determined by s w21 w22 x0 ln E1 =E2 : 3 w22 w21 For the analysis below, we de®ne the ratio E2 =E1 as the governing parameter g. To ®nd a wavefront structure nearby the waist, and, generally, how the phase behaves at the vicinity of the dark line, we have to calculate the interference ®eld as a sum of two complex amplitudes E x; z E1 x; z E2 x; z and then calculate the resulting phase U x; z: tan U x; z
Im E x; z : Re E x; z
4
Thus the equation for a wavefront family U x; z 0; 2p; 4p; . . . can be found as ImE x; z 0. A family of wavefronts (crests) is shown in Fig. 1(c) for g 0:83, w1 10, w2 100, U1 1 (here and below all ranges are normalized for k 1). First, it is seen from Fig. 1(c) that there is a lack of one wavefront inside the interval bordered by edge dislocations. Before the waist, the wavefront
51
has a ledge along the direction of propagation, after the waist the ledge is directed opposite. The ledge is smoothed far from the waist and the resulting lack of the wavefront results in a small wavelength dierence within the interval between the dislocations and outside, or phase velocity variation. The phase velocity is larger within the interval between the dislocations, the physical reason is the Gouy eect [6]: a spatially compressed light beam increases the phase velocity on the axis. To calculate the phase velocity of a 2D Gaussian beam in the waist, Eq. (1) can be applied x2 vc 1 2 2zR
1 2kzR
1
;
5
where c is the speed ofplight. There is a distance xv where v c: xv w0 = 2. When x < xv , the phase velocity exceeds c, and vice versa. Due to this phase velocity variation, wavefront being plane at the waist gradually transforms to cylindrical at the far ®eld. We can also calculate the phase velocity for the superposition of the beams vR : vR
E1 x E2 xv1 v2 ; v2 E1 x v1 E2 x
6
where vR , v1 and v2 are functions of the transverse coordinate x. For the case of constructive interference, minus sign in the numerator and denominator will be replaced by plus. The higher transversal compression of a beam results in the higher phase velocity, therefore a narrow beam with the waist parameter w1 is a little bit faster at the waist than the wide beam with the waist parameter w2 . We note, however, that the possible presence of an edge dislocation can drastically perturb the smooth velocity dependence along the transversal coordinate nearby the dislocation. Fig. 2 shows a family of equiphase lines in the cross-section XZ in the vicinity of the edge dislocation, for g 0:83. There is an additional wavefront outside the inner stripe between the dislocation lines, and a point where it enters the periodic sequence: here, at the phase saddle, lines of U x; z 0 and U x; z 2p join.
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V.A. Pas'ko et al. / Optics Communications 198 (2001) 49±56
the zero-amplitude line is localized in a very narrow region. In comparison with the wavelength, the separation between the vortex and the saddle is extremely small and amounts of about 10 2 of the wavelength. 3. Topological reactions in an interference ®eld of 2D Gaussian beams
Fig. 2. Lines of equal phase of the interference ®eld produced by two 2D Gaussian beams in the cross-section XZ nearby the edge dislocation, for g 0:83, U1 0. Saddle point is indicated as a circle, vortex as a dot. There is a p-hop of the phase at the zero-amplitude point (core of the vortex).
The equation for determination of the saddle position xs can be obtained as the coordinate of the point where a wavefront U x; z 0 splits kx2 1 kx2 1 k 2 g k 2 2zR2 2zR1 2zR1 2z R2 2 x x2 exp : 7 w21 w22 Analysis shows that the position of the saddle can be as inside the interval x0 (xs < x0 ), as outside it (xs > x0 ). First, we shall analyze the case xs < x0 which is similar to the described in Ref. [4] circular edge dislocation structure. The phase map looks identical, manifesting the vortices around zeroamplitude lines for g 0:83 (Fig. 2), and we can seek for some particular details now. The distance between vortex and saddle Dx depends on the beam parameters and can be estimated from Eq. (7) for w2 w1 as: Dx
2x20 w21 : 2k 2 w21 x0
8
The physical sense of the saddle point is the condition of the energy current stagnation: vortex ¯ow is exactly compensated by the longitudinal ¯ow, and a line of standing wave is created. Therefore the vorticity area is ended in the saddle point, and the whole torous light current around
With increasing g, x0 diminishes according to Eq. (3), and two dislocation lines come closer as well as saddles. The saddles, which are located inside the interval between the vortices, will meet at x 0 point when g zR2 2kzR1 1= zR1 2kzR2 1. This is the ®rst stage of the collapse (Fig. 3(a)). Then saddles appear again separately, but are located now symmetrical on the Z-axis (Fig. 3(b)). The maximum distance between the saddles will be reached for g 0:995 and amounts approximately to 1. In this moment the distance between vortices becomes equal to the distance between saddles. With further increase of g the saddles as well as vortices come closer. Finally, vortices and saddles collide and annihilate altogether at the point x 0 (Fig. 3(c)). With decreasing of g, saddle point moves toward the vortex and meets it when g g0 exp w21 w22 =2 w21 w22 (this condition corresponds to the coincidence of the dislocation line to the line of equality of phase velocities of the interfering waves, as will be discussed below). In the moment of this collision a saddle±vortex point appears, which is a ``pathological'' topological object. After the crossing event, the saddle appears outside the dislocation line, and thus the light ¯ow circulation around the dislocation reverses the direction. This reaction is accompanied with two similar objects creation outside the plane z 0, as Fig. 4 shows. These two new dislocations have the same ``sign'', or direction of the current circulation around them, as the initial one, so there is again a lack of one wavefront in the central part of the beam. If we start from smaller value of g, g < g0 which corresponds to the existence of three edge dislocations, and then gradually increase g until it reaches g0 value, the topological reaction will look as follows: three parallel zero-amplitude lines
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Fig. 4. Lines of equal phase in the cross-section XZ. Both wave crests (solid lines) and troughs (dashed lines) are shown, as calculated from the condition ImE x; z 0, U1 0:1. Position of vortices is shown by dots, saddles are not shown here.
the total number of critical points in the XZ crosssection is six. The signs of vortices, which are out of the waist plane (``positive''), are opposite to the sign of vortex localized in the waist plane (``negative''). After the collision, the remaining objects are one vortex with positive sign and one saddle. Of course, we discuss the events happening only in the x > 0 half plane, in the x < 0 half plane the same process happens symmetrically. Let us examine now how new edge dislocations can sprout from the saddle±vortex line. First, we have to ®nd the conditions for edge dislocation to exist out of the waist plane. There are two necessary conditions, one for destructive interference, another for the equality of the interfering waves amplitudes. The phase condition looks as follows: kx2 2R1 z
Fig. 3. Collapse of edge dislocations. (a) Equiphase lines around the dislocations which are close to the beam center, saddle points have joint. (b) Vortices approach closer, saddles are split along the Z-axis. (c) All topological objects collide and annihilate at g 1.
accompanied with three lines of stagnation (phase saddle lines) come closer until the junction. Altogether, there are three vortices and three saddles,
1 z arctan 2 zR1
kx2 1 z arctan 0: zR2 2R2 z 2 9
There is a trivial solution z 0 (the waist plane), where two initial edge dislocations are located, and a surface of destructive interference where new edge dislocations can appear 1 z2 z2R1 z2 z2R2 2 x kz z2R2 z2R1 z z arctan arctan ; 10 zR1 zR2
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natural, however, that the axial velocity is usually larger: diraction is responsible for it. Ordinary focusing cannot change this feature, and some tricks are necessary to apply for the construction of the wanted beam. As it is seen from formula (6), subtraction of a ``slow'' beam from a ``fast'' one leads to the increase of its axial velocity, and coherent addition will decrease it. Therefore we have chosen 2D analog of a Laguerre±Gaussian mode LG01 as the fast beam Fig. 5. Solid line: cross-section of the surface of destructive interference (p phase dierence between interfering beams), for x > 0. Dashed lines: cross-sections of the surfaces of the amplitude equality. For g > g0 there is only one possible crossing point and therefore one zero-amplitude line (edge dislocation). For g < g0 there are three points of the crossing and therefore three edge dislocations (saddle points are not shown here).
which can be approximated nearby the waist as x2
zR1 zR2 2z2 2z2R2 zR1 zR2 z2 R1 : k zR1 zR2 3kzR1 zR2 zR1 zR2
11
The cross-section of the surface of the destructive interference is shown in Fig. 5 by solid line. To ®nd the location of edge dislocation out of the waist plane, the surface of the amplitude equality can be found also, as shown in Fig. 5 by dashed line. Its position depends on the governing parameter g, whereas the line of the destructive interference does not depend on the amplitude ratio of the beams. This line crosses the X-axis exactly in the point which corresponds to the equality of phase velocities of the interfering waves. This coincidence is hardly occasional, and can give a hint for understanding of the vortex behavior. The consequence can be made: the vortex rotation is directed toward the area of higher phase velocity.
4. Inversion of an optical vortex circulation To prove this assumption, we have to construct somehow a beam, which possesses a pair of edge dislocations at the waist and the phase velocity between them smaller than peripheral one. It is
ELG E1
r 2 w0 4x 1 exp w w2 5 z i arctan ikz ; 2 zR
x2 kx2 i 2 w 2R z 12
and 2D Gaussian beam with the amplitude parameter E2 and the same waist parameter w0 as the slow beam. Again, the amplitude ratio g E2 =E1 will serve as the governing parameter. The 2D LG01 mode possesses increased Gouy phase shift, what insures higher phase velocity, and inherent phase defect in a form of a p step along the surfaces x2 w20 1 z2 =z2R , where amplitude ELG gets zero value. A coherent in-phase (or out of phase) addition of a 2D Gaussian beam will destroy the zero-amplitude surfaces, leaving the only two zeroamplitude lines at the waist plane, which are linear edge dislocations. Fig. 6(a) shows the result of the destructive interference of the beams. As the amplitude of the 2D LG01 mode in the waist is negative within the interval 2w0 , the subtraction of the 2D Gaussian beam will diminish the axial phase velocity here, whereas outside the interval the interference will increase the phase velocity, what is necessary for our task. The resulting wavefront family is shown in Fig. 6(b). In contrast to Fig. 1(c), an additional wavefront sheet appears within the interval bordered by edge dislocations. The detailed structure of the phase around the dislocation is shown in Fig. 7. The saddle position now is outside the interval between dislocations, and the light vortex rotates around dislocations from interior to outside the interval. Moreover, we can add the slow 2D Gaussian beam instead of the subtraction, and obtain the reversed situation.
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Fig. 7. Lines of equal phase of the destructive interference ®eld produced by 2D LG01 mode and 2D Gaussian beam in the crosssection XZ nearby the edge dislocation, for g 0:5 (compare with Fig. 2).
5. Conclusion
Fig. 6. The same as Fig. 1(b) and (c), but for the destructive interference ®eld produced by 2D LG01 mode and 2D Gaussian beam. (a) Amplitude distribution in the waist, dashed line is the resulting amplitude (negative amplitude denotes the p phase shift). (b) A wavefront sequence near the waist: there is an extra wavefront between the edge dislocations. The ledge is directed backward to the direction of the wave propagation before the waist and along the direction of propagation after the waist.
This veri®cation absolutely agrees with the expectations. A single edge dislocation can be constructed in a similar way, if we take a 2D mode r w0 x x2 kx2 exp E x; z E1 i ww w2 2R z 3 z i arctan ikz 13 2 zR as the fast beam and add/subtract a 2D Gaussian beam.
Summarizing, we have found how the transversal OV, created in an interference ®eld of two 2D Gaussian beams, transforms in topological reactions of edge dislocation collapse and ``unfolding'' [7]. The results obtained are valid as well for 3D beams. The rule for determination of the direction of a transversal OV circulation around an edge dislocation was established: the light vortex rotates toward the region of the higher phase velocity. This rule can be used for the explanation of the natural structure of phase circulation around dark rings (circular edge dislocations) observed by Karman et al. [8]. The physical reason of this feature is not clear enough and this fundamental property of light propagation seems worthwhile for further investigations.
Acknowledgements This work was supported by EOARD, Air Force Oce of Scienti®c Research and Development, Air Force Research Lab, USA (Partner Project P051).
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[5] J.F. Nye, J.V. Hajnal, J.H. Hannay, Proc. Roy. Soc. Lond. A 417 (1988) 7. [6] M. Gouy, Compt. Rendue Acad. Sci. Paris 110 (1890) 1251; A.E. Siegman, Lasers, University Science Book, Mill Valley, 1986, p. 1251, pp. 682±683. [7] J.F. Nye, J. Opt. Soc. Am. A 15 (1998) 1132. [8] G.P. Karman, M.W. Beijersbergen, A. van Duijl, J.P. Woerdman, Opt. Lett. 22 (1997) 1503; G.P. Karman, J.P. Woerdman, J. Opt. Soc. Am. A 15 (1998) 2862.