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Interference of intersecting singular beams
Optics Communications 220 (2003) 247–255 www.elsevier.com/locate/optcom
Interference of intersecting singular beams V. Pyragaite, A. Stabinis * Department of Quantum Electronics, Vilnius University, Saul_etekio 9, building 3, 2040 Vilnius, Lithuania Received 13 January 2003; received in revised form 20 March 2003; accepted 8 April 2003
Abstract A vorticity of the light field created by interference of two intersecting Laguerre–Gaussian singular beams is analysed. It is demonstrated that the number and location of the vortices present in the field depend on the propagation length as well as on the topological charges of the individual beams, their intersection angle and amplitude ratio. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.25.Hz; 42.60.Jf Keywords: Beam interference; Optical vortex
1. Introduction The vortices are spiral phase ramps around a singularity, where the phase of the beam is undetermined and its amplitude vanishes [1]. The order of the singularity multiplied by its sign is referred to as the topological charge of the vortex. A fundamental question arises about the vorticity of the light field created by superposition of individual singular beams. In the case of superposition of two coaxial Laguerre– Gaussian (LG) beams the number of existing vortices and their net topological charge are found to depend during free-space propagation on the beam relative widths and amplitudes [2]. A vorticity of the combined beam composed of two coaxial Bessel singular beams varies under diffraction [3]. The dynamical inversion of the topological charge of a superposition of LG modes carrying different charges was observed under free-space propagation [4]. The superposition of vortices nested in LG noncoaxial beams creates light patterns with a richer vortex content than that of the individual beams. The number and the location of vortices present in the field depend on amplitudes and axial separation of the individual beams [5] and vary under diffraction [6]. Our goal in this paper is to analyse a vorticity of the light field created by interference of two intersecting LG singular beams. In particular, we reveal that the number and location of the vortices present in the field
*
Corresponding author. Tel.: +370-5-236-6050; fax: +370-5-236-6006. E-mail address: algirdas.stabinis@ff.vu.lt (A. Stabinis).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01422-6
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V. Pyragaite, A. Stabinis / Optics Communications 220 (2003) 247–255
Fig. 1. Schematic depiction of the intersection of two beams.
depend on the propagation length as well as on the intersection angle of the singular beams, their topological charges and the ratio of amplitudes. In what follows, we analyse the interference of two intersecting LG beams with nested single-charged vortices (Fig. 1). In Cartesian coordinates the complex amplitude A of the resulting beam can be written as " # x þ iðy hzÞ x2 þ ðy hzÞ2 iay Aðx; y; zÞ ¼ A1 þ exp d d d2 " # 2 x þ ieðy þ hzÞ x2 þ ðy þ hzÞ iay þ A2 exp ; ð1Þ d d d2 where A1 and A2 are the complex amplitudes of the individual beams, a ¼ khd, 2h is a small intersection angle, k is a wave vector and d is a beamwidth. Eq. (1) describes an interference of two beams both carrying positively single-charged vortices at e ¼ 1 and the beams with the vortices of opposite charges at e ¼ 1. Due to symmetry we exclude from the further consideration the negative values of propagation length z. It was supposed in Eq. (1), that a diffraction of the intersecting beams is negligible. That is correct, if a size of beam overlap region d=h (see Fig. 1) is much smaller in comparison with Rayleigh range Ld ¼ kd 2 =2. So, the diffraction is unimportant when d=h Ld or a 2. We note, that Eq. (1) can be easily generalized for diffracting beams. The location of the cores of the existing vortices in the light field of two intersecting beams is determined by the complex zeroes of the equation Aðx; y; zÞ ¼ 0;
ð2Þ
see Eq. (1).
2. Interference of singular beams with equal amplitudes. Identical topological charges First, we analyse the most simple case, when the amplitudes and phases of both beams carrying positively single-charged vortices (e ¼ 1) are the same, A1 ¼ A2 . Then, for real and imaginary parts of Eq. (2) we obtain ay ay ayz ayz azd ayz x cosh cosh y sinh ¼ 0; cos sin dLd d dLd 2Ld dLd d ð3Þ ay ay ayz ayz azd ayz sinh þ y cosh ¼ 0: sin cos x sinh dLd d dLd 2Ld dLd d Obviously, there exists a central vortex corresponding to the trivial solution x ¼ 0, y ¼ 0. The charge of pffiffiffi the central vortex is positive at 0 6 z < zc ¼ ð 2Ld =aÞ and negative when z > zc . An elimination of the coordinate x 6¼ 0 in Eqs. (3) at z > 0 yields
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ay
f1 ðyÞ ¼ f2 ðyÞ; where f1 ðyÞ ¼ cos2 ; d 1 ayz azd ayz 1 cosh 2 sinh 2 þ : f2 ðyÞ ¼ 2 dLd 2yLd dLd
ð4Þ
Eq. (4) was solved numerically, and the obtained dependence yðzÞ for two values of the parameter a is presented in Figs. 2(a) and 3(a). We note, that in these cases an influence of the diffraction is rather small. Figs. 2(b) and 3(b) illustrate the graphic solution of Eq. (4) for different values of the propagation length z=Ld . At z ¼ 0 only central vortex is present in the light field. If z > 0, an appearance of the pairs of peripheral vortices is observed at minimum propagation length z, the value of which is compatible pffiffiffiwith an inequality tanhðpz=2Ld Þ < ða2 z=pLd Þ. As a result, an appearance p offfiffiffiperipheral vortices at a > p= 2 ¼ 2:22 takes place for any vanishing value z > 0. In contrary, at a < p= 2 the peripheral vortices emerge at fixed propagation length z ¼ z0 , where z0 is a solution of an equation tanhðpz0 =2Ld Þ ¼ ða2 z0 =pLd Þ. We note, that two intersecting beams leave each other at the distance z=Ld ¼ 2=a. The x coordinates of the vortex cores can be found by substitution of the calculated y values into Eqs. (3). The location of the vortex cores in x, y plane for different values of propagation length z=Ld is presented in Figs. 2(c) and 3(c). Making use of Eqs. (3) for x 6¼ 0 one obtains " 2 # x2 þ y 2 az ayz ayz : ð5Þ ¼ þ tanh 2 2 2Ld dLd dLd d
Fig. 2. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f2 (solid line) curves determine y coordinates of the peripheral vortex cores (b). z=Ld : 0 (1), 0.2 (2), 0.45 (3), 0.6 (4). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
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Fig. 3. Location of the vortex cores in the light field at a ¼ 10, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f2 (solid line) curves determine y coordinates of the peripheral vortex cores (b). z=Ld : 0 (1), 0.05 (2), 0.1 (3), 0.13 (4), 0.15 (5), 0.2 (6). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively.
At ajyjz dLd we have 2 x2 y 2 1 az þ ; 2Ld d2 d2 2
ð6Þ
and the cores of the peripheral vortices are situated on the circumference, see, for example, Fig. 3(c)(2). The typical interference fringes in the light field of two intersecting singular beams at different longitudinal positions z=Ld are presented in Fig. 4 for a ¼ 3. Now we shall discuss an influence of the beam diffraction at a < 3. A richer vortex content appears pffiffiffi in the field created by two parallel LG beams if a ratio of an axial separation and the beamwidth exceeds 2=2 [5]. In the case of two intersecting beams an axial separation hz and the beamwidth (due to diffraction) increase with a propagation distancepz.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi So, a richer vortex content should be observed under diffraction only at pffiffiffi hz=d1 > 2=2, where d1 ¼ d 1 þ z2 =L2d . Here d is a beamwidth at the waist (z ¼ 0). As a result, an obtained pffiffiffi pffiffiffi inequality is valid for a > 2. So, at a < 2 only central vortex should be present in the light field of two
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Fig. 4. Interference fringes of two intersecting singular beams at a ¼ 3 in x, y plane. z=Ld : 0 (a), 0.2 (b), 0.3 (c), 0.7 (d). Amplitude ratio A2 =A1 ¼ 1. e ¼ 1. x and y coordinates are normalized to the beamwidth d.
Fig. 5. Location of the vortex cores in the light field of two intersecting singular beams in z, y plane under diffraction. Amplitude ratio A2 =A1 ¼ 1. e ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. a: 1.4 (a), 1.5 (b), 2 (c).
intersecting beams. An appearance of the peripheral vortices is feasible at a > at a ¼ 3 an influence of the diffraction is already negligible, see Fig. 2(a).
pffiffiffi 2, see Fig. 5. We note, that
3. Interference of singular beams with equal amplitudes. Opposite topological charges Second, we analyse the interference of two singular beams of equal amplitudes (A1 ¼ A2 ) carrying singlecharged vortices of opposite charges (e ¼ 1). In this case for real and imaginary parts of Eq. (2) we find
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ay ayz azd ayz x cosh sinh y cosh ¼ 0; cos sin d dLd 2Ld dLd d ay ay ayz ayz azd ayz x sinh cosh þ y sinh ¼ 0: sin cos dLd d dLd 2Ld dLd d ayz dLd
ay
ð7Þ
An elimination of the x coordinate in Eqs. (7) at z > 0 yields 1 ayz 2yLd ayz sinh 2 þ ; f1 ðyÞ ¼ f3 ðyÞ; where f3 ðyÞ ¼ 1 cosh 2 2 dLd dLd azd
ð8Þ
see Eqs. (4). An obtained dependence yðzÞ for two values of the parameter a is presented in Figs. 6(a) and 7(a). Figs. 6(b) and 7(b) depict the graphic solution of Eq. p(8) ffiffiffi for different values of the propagation length. The function f3 ðyÞ has a minimum at y ¼ 0, if z < zc ¼ 2Ld =a, and a maximum for z > zc , see Figs. 6(b) and 7(b). We note, that in this case the vortices are absent in the light field at z ¼ 0. Their appearance is observed for any vanishing value of propagation length z > 0, Figs. 6(a) and 7(a). The coordinates x are found by substitution of the calculated y values into Eqs. (7). The location of the vortex cores in x, y plane for different values of propagation length z=Ld is presented in Figs. 6(c) and 7(c). It should be pointed out, that Eq. (5) obtained for e ¼ 1 is also compatible with Eqs. (7), which are valid for e ¼ 1. It means, that in both cases the vortex cores are located at different points of the same surface in the xyz space.
Fig. 6. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f3 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.2 (1), 0.45 (2), 0.6 (3). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
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Fig. 7. Location of the vortex cores in the light field at a ¼ 10, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f3 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.05 (1), 0.13 (2), 0.15 (3), 0.16 (4), 0.17 (5), 0.2 (6). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively.
4. Interference of singular beams with different amplitudes. Identical topological charges Third, we analyse the interference of two singular beams of different amplitudes (A1 6¼ A2 ) carrying positively single-charged vortices (e ¼ 1). For real and imaginary parts of Eq. (2) we obtain ay ayz ayz x cos exp þ b exp d dLd dLd ay azd ayz azd ayz þ sin y exp þb yþ exp ¼ 0; d 2Ld dLd 2Ld dLd ð9Þ ay ayz ayz exp b exp x sin d dLd dLd ay azd ayz azd ayz þ cos y exp þb yþ exp ¼ 0; d 2Ld dLd 2Ld dLd
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Fig. 8. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f4 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.03 (1), 0.2 (2), 0.45 (3), 0.6 (4). Amplitude ratio A2 =A1 ¼ 0:9. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
where b ¼ A2 =A1 . An elimination of the coordinate x in Eqs. (9) gives f1 ðyÞ ¼ f4 ðyÞ; where ( 2 ), ayz ayz azd ayz ayz 2 f4 ðyÞ ¼ y exp exp 2 bexp b exp 2 ð4byÞ: dLd dLd 2Ld dLd dLd
ð10Þ
The numerically calculated dependence yðzÞ for a ¼ 3 and b ¼ 0:9 is presented in Fig. 8(a). Fig. 8(b) illustrates the graphic solution of Eq. (10) for different values of propagation length z=Ld . We note, that at y ! 0 follows f4 ðyÞ ðazd=8Ld byÞð1 b2 Þ. In this case a vortex core located initially at x ¼ y ¼ 0 is shifted from its central position during the beam propagation. The pairs of the vortices with the opposite charges appear at propagation length z ¼ z0 , where z0 is a solution of an equation ðz0 a2 =pLd Þ ¼ ½expðpz0 =Ld Þ c= ½expðpz0 =Ld Þ þ c, where c ¼ b for y > 0 and c ¼ 1=b at y < 0. The coordinates x of the vortex cores can be calculated by substitution of the y values into Eqs. (9). The location of the vortex cores in x, y plane for the different values of propagation length z=Ld is presented in Fig. 8(c). 5. Conclusions The vorticity of the field created by interference of two intersecting singular beams was analysed. In the case of larger intersection angle (smaller overlap region of two beams) the obtained light pattern possesses much richer vortex content than that given by the arithmetics of the topological charges of the individual beams. It was shown that the number and location of the vortices present in the field depend on the propagation length as well as on the topological charges of the individual beams, their intersection angle and amplitude ratio.
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In the case of smaller intersection angle of two singular beams (larger overlap region) the diffraction essentially simplifies an interference pattern.
References [1] [2] [3] [4] [5] [6]
J.F. Nye, M.V. Berry, Proc. R. Soc. A 336 (1974) 165. M.S. Soskin, V.N. Gorshkov, M.V. Vasnetsov, J.T. Malos, N.R. Heckenberg, Phys. Rev. A 56 (1997) 4064. S. Orlov, K. Regelskis, V. Smilgevicius, A. Stabinis, Opt. Commun. 209 (2002) 155. G. Molina-Terriza, J. Recolons, J.P. Torres, L. Torner, E. Wright, Phys. Rev. Lett. 87 (2001) 023902. G. Molina-Terriza, J. Recolons, L. Torner, Opt. Lett. 25 (2000) 1135. V. Pyragaite, A. Stabinis, Opt. Commun. 213 (2002) 187.
Interference of intersecting singular beams V. Pyragaite, A. Stabinis * Department of Quantum Electronics, Vilnius University, Saul_etekio 9, building 3, 2040 Vilnius, Lithuania Received 13 January 2003; received in revised form 20 March 2003; accepted 8 April 2003
Abstract A vorticity of the light field created by interference of two intersecting Laguerre–Gaussian singular beams is analysed. It is demonstrated that the number and location of the vortices present in the field depend on the propagation length as well as on the topological charges of the individual beams, their intersection angle and amplitude ratio. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.25.Hz; 42.60.Jf Keywords: Beam interference; Optical vortex
1. Introduction The vortices are spiral phase ramps around a singularity, where the phase of the beam is undetermined and its amplitude vanishes [1]. The order of the singularity multiplied by its sign is referred to as the topological charge of the vortex. A fundamental question arises about the vorticity of the light field created by superposition of individual singular beams. In the case of superposition of two coaxial Laguerre– Gaussian (LG) beams the number of existing vortices and their net topological charge are found to depend during free-space propagation on the beam relative widths and amplitudes [2]. A vorticity of the combined beam composed of two coaxial Bessel singular beams varies under diffraction [3]. The dynamical inversion of the topological charge of a superposition of LG modes carrying different charges was observed under free-space propagation [4]. The superposition of vortices nested in LG noncoaxial beams creates light patterns with a richer vortex content than that of the individual beams. The number and the location of vortices present in the field depend on amplitudes and axial separation of the individual beams [5] and vary under diffraction [6]. Our goal in this paper is to analyse a vorticity of the light field created by interference of two intersecting LG singular beams. In particular, we reveal that the number and location of the vortices present in the field
*
Corresponding author. Tel.: +370-5-236-6050; fax: +370-5-236-6006. E-mail address: algirdas.stabinis@ff.vu.lt (A. Stabinis).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01422-6
248
V. Pyragaite, A. Stabinis / Optics Communications 220 (2003) 247–255
Fig. 1. Schematic depiction of the intersection of two beams.
depend on the propagation length as well as on the intersection angle of the singular beams, their topological charges and the ratio of amplitudes. In what follows, we analyse the interference of two intersecting LG beams with nested single-charged vortices (Fig. 1). In Cartesian coordinates the complex amplitude A of the resulting beam can be written as " # x þ iðy hzÞ x2 þ ðy hzÞ2 iay Aðx; y; zÞ ¼ A1 þ exp d d d2 " # 2 x þ ieðy þ hzÞ x2 þ ðy þ hzÞ iay þ A2 exp ; ð1Þ d d d2 where A1 and A2 are the complex amplitudes of the individual beams, a ¼ khd, 2h is a small intersection angle, k is a wave vector and d is a beamwidth. Eq. (1) describes an interference of two beams both carrying positively single-charged vortices at e ¼ 1 and the beams with the vortices of opposite charges at e ¼ 1. Due to symmetry we exclude from the further consideration the negative values of propagation length z. It was supposed in Eq. (1), that a diffraction of the intersecting beams is negligible. That is correct, if a size of beam overlap region d=h (see Fig. 1) is much smaller in comparison with Rayleigh range Ld ¼ kd 2 =2. So, the diffraction is unimportant when d=h Ld or a 2. We note, that Eq. (1) can be easily generalized for diffracting beams. The location of the cores of the existing vortices in the light field of two intersecting beams is determined by the complex zeroes of the equation Aðx; y; zÞ ¼ 0;
ð2Þ
see Eq. (1).
2. Interference of singular beams with equal amplitudes. Identical topological charges First, we analyse the most simple case, when the amplitudes and phases of both beams carrying positively single-charged vortices (e ¼ 1) are the same, A1 ¼ A2 . Then, for real and imaginary parts of Eq. (2) we obtain ay ay ayz ayz azd ayz x cosh cosh y sinh ¼ 0; cos sin dLd d dLd 2Ld dLd d ð3Þ ay ay ayz ayz azd ayz sinh þ y cosh ¼ 0: sin cos x sinh dLd d dLd 2Ld dLd d Obviously, there exists a central vortex corresponding to the trivial solution x ¼ 0, y ¼ 0. The charge of pffiffiffi the central vortex is positive at 0 6 z < zc ¼ ð 2Ld =aÞ and negative when z > zc . An elimination of the coordinate x 6¼ 0 in Eqs. (3) at z > 0 yields
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249
ay
f1 ðyÞ ¼ f2 ðyÞ; where f1 ðyÞ ¼ cos2 ; d 1 ayz azd ayz 1 cosh 2 sinh 2 þ : f2 ðyÞ ¼ 2 dLd 2yLd dLd
ð4Þ
Eq. (4) was solved numerically, and the obtained dependence yðzÞ for two values of the parameter a is presented in Figs. 2(a) and 3(a). We note, that in these cases an influence of the diffraction is rather small. Figs. 2(b) and 3(b) illustrate the graphic solution of Eq. (4) for different values of the propagation length z=Ld . At z ¼ 0 only central vortex is present in the light field. If z > 0, an appearance of the pairs of peripheral vortices is observed at minimum propagation length z, the value of which is compatible pffiffiffiwith an inequality tanhðpz=2Ld Þ < ða2 z=pLd Þ. As a result, an appearance p offfiffiffiperipheral vortices at a > p= 2 ¼ 2:22 takes place for any vanishing value z > 0. In contrary, at a < p= 2 the peripheral vortices emerge at fixed propagation length z ¼ z0 , where z0 is a solution of an equation tanhðpz0 =2Ld Þ ¼ ða2 z0 =pLd Þ. We note, that two intersecting beams leave each other at the distance z=Ld ¼ 2=a. The x coordinates of the vortex cores can be found by substitution of the calculated y values into Eqs. (3). The location of the vortex cores in x, y plane for different values of propagation length z=Ld is presented in Figs. 2(c) and 3(c). Making use of Eqs. (3) for x 6¼ 0 one obtains " 2 # x2 þ y 2 az ayz ayz : ð5Þ ¼ þ tanh 2 2 2Ld dLd dLd d
Fig. 2. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f2 (solid line) curves determine y coordinates of the peripheral vortex cores (b). z=Ld : 0 (1), 0.2 (2), 0.45 (3), 0.6 (4). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
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Fig. 3. Location of the vortex cores in the light field at a ¼ 10, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f2 (solid line) curves determine y coordinates of the peripheral vortex cores (b). z=Ld : 0 (1), 0.05 (2), 0.1 (3), 0.13 (4), 0.15 (5), 0.2 (6). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively.
At ajyjz dLd we have 2 x2 y 2 1 az þ ; 2Ld d2 d2 2
ð6Þ
and the cores of the peripheral vortices are situated on the circumference, see, for example, Fig. 3(c)(2). The typical interference fringes in the light field of two intersecting singular beams at different longitudinal positions z=Ld are presented in Fig. 4 for a ¼ 3. Now we shall discuss an influence of the beam diffraction at a < 3. A richer vortex content appears pffiffiffi in the field created by two parallel LG beams if a ratio of an axial separation and the beamwidth exceeds 2=2 [5]. In the case of two intersecting beams an axial separation hz and the beamwidth (due to diffraction) increase with a propagation distancepz.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi So, a richer vortex content should be observed under diffraction only at pffiffiffi hz=d1 > 2=2, where d1 ¼ d 1 þ z2 =L2d . Here d is a beamwidth at the waist (z ¼ 0). As a result, an obtained pffiffiffi pffiffiffi inequality is valid for a > 2. So, at a < 2 only central vortex should be present in the light field of two
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Fig. 4. Interference fringes of two intersecting singular beams at a ¼ 3 in x, y plane. z=Ld : 0 (a), 0.2 (b), 0.3 (c), 0.7 (d). Amplitude ratio A2 =A1 ¼ 1. e ¼ 1. x and y coordinates are normalized to the beamwidth d.
Fig. 5. Location of the vortex cores in the light field of two intersecting singular beams in z, y plane under diffraction. Amplitude ratio A2 =A1 ¼ 1. e ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. a: 1.4 (a), 1.5 (b), 2 (c).
intersecting beams. An appearance of the peripheral vortices is feasible at a > at a ¼ 3 an influence of the diffraction is already negligible, see Fig. 2(a).
pffiffiffi 2, see Fig. 5. We note, that
3. Interference of singular beams with equal amplitudes. Opposite topological charges Second, we analyse the interference of two singular beams of equal amplitudes (A1 ¼ A2 ) carrying singlecharged vortices of opposite charges (e ¼ 1). In this case for real and imaginary parts of Eq. (2) we find
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ay ayz azd ayz x cosh sinh y cosh ¼ 0; cos sin d dLd 2Ld dLd d ay ay ayz ayz azd ayz x sinh cosh þ y sinh ¼ 0: sin cos dLd d dLd 2Ld dLd d ayz dLd
ay
ð7Þ
An elimination of the x coordinate in Eqs. (7) at z > 0 yields 1 ayz 2yLd ayz sinh 2 þ ; f1 ðyÞ ¼ f3 ðyÞ; where f3 ðyÞ ¼ 1 cosh 2 2 dLd dLd azd
ð8Þ
see Eqs. (4). An obtained dependence yðzÞ for two values of the parameter a is presented in Figs. 6(a) and 7(a). Figs. 6(b) and 7(b) depict the graphic solution of Eq. p(8) ffiffiffi for different values of the propagation length. The function f3 ðyÞ has a minimum at y ¼ 0, if z < zc ¼ 2Ld =a, and a maximum for z > zc , see Figs. 6(b) and 7(b). We note, that in this case the vortices are absent in the light field at z ¼ 0. Their appearance is observed for any vanishing value of propagation length z > 0, Figs. 6(a) and 7(a). The coordinates x are found by substitution of the calculated y values into Eqs. (7). The location of the vortex cores in x, y plane for different values of propagation length z=Ld is presented in Figs. 6(c) and 7(c). It should be pointed out, that Eq. (5) obtained for e ¼ 1 is also compatible with Eqs. (7), which are valid for e ¼ 1. It means, that in both cases the vortex cores are located at different points of the same surface in the xyz space.
Fig. 6. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f3 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.2 (1), 0.45 (2), 0.6 (3). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
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Fig. 7. Location of the vortex cores in the light field at a ¼ 10, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f3 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.05 (1), 0.13 (2), 0.15 (3), 0.16 (4), 0.17 (5), 0.2 (6). Amplitude ratio A2 =A1 ¼ 1. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively.
4. Interference of singular beams with different amplitudes. Identical topological charges Third, we analyse the interference of two singular beams of different amplitudes (A1 6¼ A2 ) carrying positively single-charged vortices (e ¼ 1). For real and imaginary parts of Eq. (2) we obtain ay ayz ayz x cos exp þ b exp d dLd dLd ay azd ayz azd ayz þ sin y exp þb yþ exp ¼ 0; d 2Ld dLd 2Ld dLd ð9Þ ay ayz ayz exp b exp x sin d dLd dLd ay azd ayz azd ayz þ cos y exp þb yþ exp ¼ 0; d 2Ld dLd 2Ld dLd
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Fig. 8. Location of the vortex cores in the light field at a ¼ 3, e ¼ 1 in z, y (a) and x, y (c) planes. Intersection points of f1 (dotted line) and f4 (solid line) curves determine y coordinates of the vortex cores (b). z=Ld : 0.03 (1), 0.2 (2), 0.45 (3), 0.6 (4). Amplitude ratio A2 =A1 ¼ 0:9. Filled and open circles mark the cores of positively and negatively single-charged vortices, respectively. Solid lines in (a) location of the vortex cores under diffraction.
where b ¼ A2 =A1 . An elimination of the coordinate x in Eqs. (9) gives f1 ðyÞ ¼ f4 ðyÞ; where ( 2 ), ayz ayz azd ayz ayz 2 f4 ðyÞ ¼ y exp exp 2 bexp b exp 2 ð4byÞ: dLd dLd 2Ld dLd dLd
ð10Þ
The numerically calculated dependence yðzÞ for a ¼ 3 and b ¼ 0:9 is presented in Fig. 8(a). Fig. 8(b) illustrates the graphic solution of Eq. (10) for different values of propagation length z=Ld . We note, that at y ! 0 follows f4 ðyÞ ðazd=8Ld byÞð1 b2 Þ. In this case a vortex core located initially at x ¼ y ¼ 0 is shifted from its central position during the beam propagation. The pairs of the vortices with the opposite charges appear at propagation length z ¼ z0 , where z0 is a solution of an equation ðz0 a2 =pLd Þ ¼ ½expðpz0 =Ld Þ c= ½expðpz0 =Ld Þ þ c, where c ¼ b for y > 0 and c ¼ 1=b at y < 0. The coordinates x of the vortex cores can be calculated by substitution of the y values into Eqs. (9). The location of the vortex cores in x, y plane for the different values of propagation length z=Ld is presented in Fig. 8(c). 5. Conclusions The vorticity of the field created by interference of two intersecting singular beams was analysed. In the case of larger intersection angle (smaller overlap region of two beams) the obtained light pattern possesses much richer vortex content than that given by the arithmetics of the topological charges of the individual beams. It was shown that the number and location of the vortices present in the field depend on the propagation length as well as on the topological charges of the individual beams, their intersection angle and amplitude ratio.
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In the case of smaller intersection angle of two singular beams (larger overlap region) the diffraction essentially simplifies an interference pattern.
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